Sentence Correction

The author of a book on the women of the Temperance Movement suggests that the single trait common to the most influential members was a vision of her public work like an extension of the safe-guarding of moral integrity in the private household, which was considered the responsibility of wives and mothers.

(A) her public work like an extension of the safe-guarding
(B) their public work as an extension of the safe-guarding
(C) public work for her as an extension of the safe-guarding
(D) their public work like an extension of the safe-guarding
(E) her public work as an extension of the safe-guarding

My choice is A because I see that public work is singular, so the pronoun should be singular, only "her" is appropriate.

Furthermore, usage of like vs as: I do not understand the role of as/like here. OE stated that "as" is idiomatically correct in the phrase "as an extension". Is "as an extension" idiom, isn't it?

Can anyone help me explain.

Thanks. OA later some discussion.

Source: gmatclub test