I see here and in somewhat different terms here that if the index of $Z(G) = 4$, $G/Z(G)$ cannot be cyclic.
Forgive me if the question should be reworded, as I may be asking more along the lines of how to prove $G/Z(G)$ cannot be cyclic rather than the main question, but given this idea seems implied in previous answers I've seen, I cannot understand the proofs I've found.
Thank you.