All four are correct. "During the last summer" is a bit uncommon in this particular use. " "During the previous summer" would work.
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AnonymousIt was the summer your mother was staying with us. ...
It was the summer your mother stayed with us.Both good.AnonymousBoth good as shoDuring theLast summer we stayed in a beautiful hotel in Spain. Would it also be correct to sayDuring theLast summer we were staying in a beautiful hotel in Spain.