Find a normal subgroup of v which is not normal in a4.

he elements of the Klein 44 - group sitting inside A4 A4 are precisely the identity, and all elements of A4 A4 of the form (ij) (kℓ) (ij) (kℓ) (the product of two disjoint transpositions).

Since conjugation in Sn Sn (and therefore in An An) does not change the cycle structure, it follows that this subgroup is a union of conjugacy classes, and therefore is normal.