Summer officially begins at the solstice on 21st/22nd June and ends at the autumnal equinox on 21st/22nd September. On that basis, Midsummer would be around the beginning of August! However, it is obvious that standard phrases such as "flaming June" and "hotter than a June bride in a feather bed" have nothing whatever to do with springtime phenomena, which that official definition would largely mean.
As the second definition of �summer', The Oxford English Dictionary says:
"in popular use, comprising in the northern hemisphere the period from mid-May to mid-August."
Traditionally, Midsummer's Day was June 24th and - in a temperate climate such as ours - the seasons are all about the same length...ie 13 weeks or so. Accordingly, summer would, in most people's view, begin some 6� weeks before June 24th, which is around May 10th , and end around August 9th. That fits almost perfectly with what the OED says.
So, you can take your pick...the official definition or the one that common language and experience tells you is so.
(Bear in mind that, if June 22nd is Day 1 of summer and June 24th is Midsummer's Day, then June 27th must be the first day of Autumn!!)