For those who are having difficulty maintaining a certain number of push-ups in each set, such as doing 15 push-ups each in 5 sets, when you are doing your push-up, do not be in a rush to complete your push-up. Always take a minimum of 20 minutes to rest before you can start another set so that you will have enough strength to do the same number of push-ups you did in your first set.
I think 20mins is too much to take a rest, when you feel that tiredness while doing the push up and you don't stop that's when you get the effects more and 30secs is enough for a rest. Whenever you rest for long like up to 20mins like you said, it might discourage you from continuing the task for the day, for me I do 100 every morning with some other fitness exercises, 30secs rest and I go again, like 2 to 3 times repeating same process even at night and whenever you do your exercise take a lot of water, it helps.