Let $K \subset L$ be a finite Galois extension, $\sigma$ an automorphism of $L$ (not necessarily fixing $K$) and let $E$ be a finite-dimensional vector space over $L$ together with an $\sigma$-linear endomorphism $\phi : E \rightarrow E$. I would like to do Galois descent. So my question is "Under what conditions does $\phi$ ascent from a $\sigma$-linear endomorphism defined over $K$?".

Thanks.

EDIT: I guess I should describe my problem more precisely.

$k$ a perfect field of char. $p>0$, $K$ the fraction field of the Witt ring $W(k)$, $F$ finite extension of $\mathbb{Q}_p$, $L$ the compositum of $K$ and $F$ in $\bar{K}$, $\tau$ the Frobenius automorphism of $L$ over $F$ and $\sigma$ the Frobenius automorphism of $K$.

A $\tau$-$L$-space is a finite dimensional vector space $V$ over $L$ together with a $\tau$-semilinear bijection $\Phi: V\rightarrow V$.

If I'm given an $\sigma^f$-isocrystal $V$ over $k$, then I can associate to it an $\tau$-$L$-space as follows. Let $F$ be the unique unramified extension of $\mathbb{Q}_p$ of degree $f$, $L=KF$ then $\tau|_K = \sigma^f$ and tensoring with $L$ gives a $\tau$-$L$-space.

I'm interested in the other direction. Let $F$ be an unramified extension of $\mathbb{Q}_p$ of degree $f$ and $E$ an $\tau$-$L$-space. Does $E$ come from a $\sigma^f$-isocrystal over $k$?