Core aerating and seeding go hand-in-hand as second to preferable seeding methods. The reason is the exposed cores provide an ideal condition for seed/soil contact. Plus, as the cores mulch themselves back into the soil, they carry the seeds with them. Is that what you're asking? The growing seasons, which are the periods of active growth, is when you want to core aerate. So, your timing is ideal. The only thing I would suggest is beginning your process as soon as possible. Rather than waiting until the end of the month, make it mid-August if you can. I say because this is also the season to battle Poa, and just killing it off and overseeding is not enough. After the new grass is established and has been mowed 2-3 times, apply a pre-emergent herbicide that is labeled for controlling Poa annua, like Balan, Dimension, or other (whichever also offers post-emergence effectiveness) or something like Prograss (very, very expensive) as soon as possible. Check herbicide label for exactly when application after seeding can take place. Incidentally, core aerate and overseed the whole the lawn, not just certain areas. Here is irrigation schedule for the seeds water 15-20 minutes twice a day for two weeks water 20-30 minutes once a day for one week water 30-45 minutes once a day every other day for one week water 30-45 minutes once a day twice a week for one week move into deep irrigation, increasing the time to provide 1 inch of water all over and decreasing the frequency to just once a week. Starting off, the schedule supplies roughly 1/4 inch of water, then increases that amount while decreasing frequency of application at the same time. Like practically everything that concerns lawn care, this schedule is a general guideline and should be modified to accomodate your specific conditions. The lengths of time should be modified if you have an automatic sprinkler system since that will not take as long to provide adequate moisture. So, decrease amount of water (time) but maintain frequency as is. The tuna cans test is recommended. If it is still especially warm, you may want to irrigate 3 times a day (10-15 minutes if necessary) for that first couple weeks. Your objective is to keep the upper 1 inch of soil moist and not let the seeds dry out. Baby the grass for the rest of year. Once the seeds begin to sprout, don't walk on it if at all possible.

You really don't need either. Cut the lawn as short as you can the day of seeding, remove the clippings, then use a rake to remove as much as the thatch as possible. Goal is seed to soil contact.Then use a broadcast spreader to spread the seed. A verticutter/power rake can make the job of thatch removal easier, but depending on the size of your lawn, it may not be worth the trouble,

I don't think you need to aerate it. If your soil is hard, the solution is to apply shampoo at 3 ounces per 1,000 square feet and water that in as described below. If you have thin KBG in a shady area, then overseed with a fine bladed fescue that will tolerate the shade. Otherwise good lawn care should help. What is good lawn care? The following is in order of importance with 1 being much more important than 2 or 3. Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall. It is not uncommon on this site for someone to write in to say that simply fixing their watering was all they needed and everything cleared up. Mowing. Mulch mow KBG at the mower's highest setting every week during the growing season. Fertilizer. Fertilize on Memorial Day, Labor Day, and about Thanksgiving depending on the grass. The last one for the year should be after the grass stops growing, but before it turns brown. The last one should be a fast release, high N fertilizer. I think you get more bang for your buck by using organic fertilizer like soybean meal, Milorganite, corn meal, alfalfa pellets (rabbit chow), or corn gluten meal. Apply any of those at 15 pounds per 1,000 square feet as often as you can afford to (in addition to the above 3 dates). The NPK numbers are lower for organics than for chemical fertilizers, but the goodness of the material is not measured in terms of NPK. Organic fertilizers are absolutely no hassle to use. You can't make a mistake and burn the grass. It doesn't wash away except under extraordinary circumstances (like flooding). You cannot over apply unless you smother the grass with it (about 70 pounds per 1,000 square feet will do that). Organics don't need to be watered in. And organics work as shown in this picture. That picture was sent in by mrmumbles a few years ago. He dropped a couple handfuls of alfalfa pellets in mid May and took this picture in mid June. You can see the improved color, density, and growth. And here's a picture of morpheuspa's KBG lawn where he practices deep and infrequent watering. The neighbors water every day. Picture was taken in July several years ago.

Awwww jeeze! You hit most of my pet peeves. Had you written in here before going to all that trouble I might have been able to talk you out of everything you did. There is an easier way to soften your soil than core aeration. Spray shampoo at a rate of 3 ounces per 1,000 square feet works over the entire surface rather than just the few hundred holes you punched in it. Overseeding bermuda is never necessary. If you have thin spots in bermuda the reason is usually too much shade. There is no solution other than take out the bermuda and put in flowers. Further, now is a bad time to seed bermuda. It is too cool. June or July is a much better time. Top dressing with city compost is a lot of work that could be avoided. This part could be worse, but I don't think it's needed.