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