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The strong isomorphism classes of extensions of finite groups areparametrized by orbits of a prescribed action on the second cohomology group.We study these orbits in the case of extensions of a finite abelian $p$-groupby a cyclic factor of order $p$. As an application, we compute number and sizesof these orbits when the initial $p$-group is generated by at most $3$elements.

Orbits classifying extensions of prime power order groups