If I were you, I'd do some photo research on YOUTUBE, googling for videos of Agia Marina, Marathona etc. In my only beach trip, I found the Ag. Marina beach too narrow, and crowded even at the last of May. Perhaps Marathona or other beaches will have more room ... all of them will have places to eat. Here is one place that shows its cafe and sunbeds -- https://www.aeginagreece.com/aegina-island/directory/aegina-listing/restaurants-cafes-bars/beach-bar/la-palma-marathonas/
This link, toa beach restaurant on Agia Marina, shows how very Narrow the beach is by comparison
... and in July the situation will be maximized.
I'm surprised that you say you 've always gone there, but you are asking others to recommend.? Also, it isn't helpful to say only "group" without indicating whether you mean 5-6 friends or 30 schoolchildren. It DOES make a difference. Are you concerned about the cost of transporting them around the island?? we cannot help if you don't share more.
I've found this CANDID website helpful http://www.greekisland.co.uk/aegina/aegina.htm when advising people about Aegina. There's also a terrific photo album of a one-day visit to Aegina in June - https://www.flickr.com/photos/36264706@N03/sets/72157621604646139/detail/ -- this wonderful pic sequence is totally convincing evidence why you should look ahead (gtp.gr) and take your group on a CONVENTIONAL -- open deck -- ferry, rather than on a hydrofoil (no decks inclosed, no view at all).
Good luck, come back with more facts re group size & transport concerns/plans and we could be more helpful