C. John suggested helping Jane with the washing-up.
Consider some contexts in which this sentence might occur.
Julie asked John what she could do to speed things up. John suggested helping Jane with the washing-up.
Jane pointed out that Julie was old enough now to have some regular chores, and asked John if he had any ideas. John suggested helping Jane with the washing-up.
suggest VERBing does not imply that VERB has any particular subject; specifically, it does not imply that the subject suggests that he should VERB. Consequently, this sentence does not, as the quotation does, imply that John made an offer to help; it at most puts helping Jane forward as a good idea.
