@Marty Isaacs: I appreciate your input. Indeed, the situation with the additive group Z of integers is quite clear: if K_n=m_n Z where m_n is in Z then all terms K_n can be simultaneously supplemented if and only if there is integer k which is coprime to all m_n. F.e. 2Z+3^nZ = Z for all natural numbers n > 1. Information on other, less well-behaving, classes of groups might be more intriguing, though.

To make it meaningful: let $V$ be a vector space, $\varphi$ a linear operator of $V$ and $b$ a fixed vector of $V.$ Find the full preimage of $b$ under $\varphi.$ This works for all dimensions, finite or infinite.

Thank you very much indeed. What is required of Aut(Fnk(Fn)) to guarantee existence of an example in question seems to be probable.