να ανταποκριθούν is actually "to respond by acting, by doing something" You mean να αποκριθούν=to respond, i.e. to reply
They answer = αυτοί απαντούν _
They reply, respond to a question = αυτοί αποκρίνονται _
They respond e.g. to a fundraising initiative = αυτοί ανταποκρίνονται
I am not sure I understand the difference between 1) αυτοί αποκρίνονται and 2) αυτοί απαντούν _
I understand that 1) is to reply to question/answer the question but 2) answer what? give you give some examples?
To be honest. there is no significant difference between those two. Αποκρίνομαι translates to respond, usually refers to someone returning communication and is a bit less common to use, while απαντώ translates to answer, and it refers to someone giving the "solution" to a problem or question, through the answer. However, this difference is more significant in English than it os in Greek, since απαντώ is used for almost everything. ^.^