Is there a word for answering a question with a question?

I am aware that answering student questions with further, leading questions is sometimes dubbed “Socratic,” but I am asking more broadly about all occasions where someone asks a question and, instead of an answer, receives another question in response. Has a word ever been coined to name this phenomenon?

This method of answering questions with questions, in order to let the questioner realize that he can find the answer by reasoning (Socrates would say that the answer was in him all along), is called maieutics (the related adjective being maieutic).

Such a question can be called a counter-question, but I do not believe there is an English word for actually posing such a question. You could make one up, such as "counter-questioning". That should be understood, at least.

Is it not true that counter-question is a good match that describes the exact structure that you inquire about? Can you have a counter-question without an initial question?

Also, wouldn't you agree, though it might be obvious, that the second question (the answering question) is called a rhetorical question?

Out of numerous figures that are are related to this type of address, if I use for example interrogatio and question my own answer, preferably with more style than I employ, am I not actually confirming and reinforcing the answer that I have given?

Other figures are: erotema, anacoenosis, anthypophora, dianoea, aporia, epiplexis, exuscitatio, pysma and ratiocinatio and some of them cover exactly the meaning that you mention in comments: challenging the initial question.

In case that the second question is not a rhetorical question, but a real question that is raised by the first question then I would say you are simply investigating the subject in search for stasis (and the term counter-question still covers it).