Imbedding problem with non-Abelian kernel of order \(p^ 4\) (Q5896589)

Let p be a prime and let A be the group of order \(p^ 4\) with generators a and b and relations \(a^{p^ 2}=(a,b)\), \(b^ p=1\). The author considers the imbedding problem for a p-extension of algebraic number fields with kernel A. He shows that this imbedding problem is solvable if and only if the compatibility condition is fulfilled and the corresponding problems for infinite primes are solvable.