I did this last year with friends and got it tweaked over time so that we felt we were getting maximum gains without taking up too much time or getting too exhausted...
We were all in decent shape at the start, so 50 pushups per set was how we started. We did 50 pushups, 4 times a day Monday through Friday, at random times, wherever it was convenient. I even did some in a bank bathroom once...
We rested weekends, and every two weeks or so retested or max. My max starting out was around 75, and by the end of the 3 months I did 108.
We just started again this year, I'm in a lot worse shape than last year, so am only doing 40 per set... for now!
Go go go!!!