Some \(p\)-groups with two generators which satisfy certain conditions arising from arithmetic in imaginary quadratic fields (Q1204168)

Let \(p\) be an odd prime. We give a list of certain types of \(p\)-groups \(G\) with two generators which satisfy the following two conditions (A) and (B): (A) \([\text{Ker }V_{G\to H}: [G,G]]=[G:H]\) for the transfer homomorphism \(V_{G\to H}: G\to H/[H,H]\) of \(G\) to every normal subgroup \(H\) with cyclic quotient \(G/H\), and (B) there exists an automorphism \(\varphi\) of \(G\) of order 2 such that \(g^{\varphi+1}\in[G,G]\) for every \(g\in G\). These conditions are necessary for \(G\) to be the Galois group of the second \(p\)-class field of an imaginary quadratic field. The list contains such a group that may be useful to find an imaginary quadratic field with an interesting property on the capitulation problem.