On some realizable metabelian $5$-groups

Let $G$ be a $5$-group of maximal class and $γ_2(G) = [G, G]$ its derived group. Assume that the abelianization $G/γ_2(G)$ is of type $(5, 5)$ and the transfers $V_{H_1\to γ_2(G)}$ and $V_{H_2\to γ_2(G)}$ are trivial, where $H_1$ and $H_2$ are two maximal normal subgroups of $G$. Then $G$ is completely determined with the isomorphism class groups of maximal class. Moreover the group $G$ is realizable with some fields $k$, which is the normal closure of a pure quintic field.

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