  Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an
exact functor from torsion-free modules over the endomorphism ring ${\rm
End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties
over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show
that every elliptic curve with complex multiplication geometrically is
isogenous over the ground field to one with complex multiplication by a maximal
order.
