This lot should help.
Spring came to refer, in the 16th century, to "the season of new growth", taken from the phrase "spring of the year". That was a reference to the notion of new growth bursting forth (imagine those time-lapse videos showing plants growing and flowers opening. You get the idea) after winter. Prior to that, in Old English, spring referred to "water rising from the ground" and "the source or head of a stream or river". Today it is the former meaning which remains. The sense in all of these uses is "rapid movement", and the Indo-European root with that meaning is sprengh-. Other Germanic languages have similar words: German and Dutch springen means "jump", just like the English verb form of the word, though Swedish springa retains more of the "rapid movement" sense as it means "run".
Summer comes to us from the Middle English sumer, from Old English sumor; akin to Old High German & Old Norse sumer summer, Sanskrit samA year, season
Autumn
This word came to English from Old French autompne "autumn", and the French got it from Latin autumnus.
Winter
This word comes to us from Old English and has had the same form since that time, with some variations in spelling here and there (the earliest recorded example of the word has it as wintra). There are cognates in the Germanic languages: Old Frisian and Dutch winter, Old High German wintar, Old Norse vetr, and Gothic wintrus. It is thought that these all derive from a nasalized form of the Indo-European root *wed- "to be wet", a reference to the winter rain and snow so common to northwestern Europe. This would make winter related to wet, water, and even otter.