'in summer' wouldn't be a specific summer, but rather something that happens in any summer. This would translate to 'om sommeren'.
'in the summer' would specifically be the upcoming summer, which would translate to 'i sommer' or 'til sommeren'. ('i sommer' can also refer to the previous summer, but the verb would hav to be inflected in the past tense.)
'på sommer' is not correct in Norwegian.