> 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.