It is proved that, if G is a finite group with a nontrivial normal 2-subgroup Q such that G/Q A7 and an element of order 5 from G acts without fixed points on Q, then the extension of G by Q is splittable, Q is an elementary abelian group, and Q is the direct product of minimal normal subgroups of G each of which is isomorphic, as a G/Q-module, to one of the two 4-dimensional irreducible GF(2)A7-modules that are conjugate with respect to an outer automorphism of the group A7.