C is awkward. He suggested helping Jane, to whom? – Damkerng T. 4 hours ago
I agree with @DamkerngT. Answer C does a good job of capturing the meaning of> >the original quote, but it's not written in good English. It should say >something like, "John suggested that he could help Jane with the washing-up." – >J.R.♦ 4 hours ago"
Actually, it's perfectly correct English, although the use of the past tense verb with a present participle is less common in modern use; also, there need not be a person to whom he needs to suggest anything - sentences do not require an object, only a subject.
suggest VERBing does not imply any particular subject to VERB; it merely puts >VERBing forward as a good idea. Consequently, this sentence does not, as the >quotation does, imply that John made an offer to help.
It doesn't need to imply anything, because it states it pretty explicitly in that sentence, by using the pronoun, "Jane".
He VERBED Jane. ahem Not what I meant. And you don't "verb" a subject, anyway, subjects verb objects, but - as mentioned above - you don't need to explicitly state the object - a sentence need only have a subject and a finite verb in order to be complete.
Importantly, "helping" isn't the finite verb in that sentence - "suggested" is.