Identity relating Frobenius to multiplication by p on the ring of Witt vectors

Question

Let $k$ be a field of characteristic $p$, $F$ be the Frobenius map of Witt vectors, and $V$ the transfer map on $W(k)$. I'm trying to show that $FV(a) = pa$ where $a$ is a Witt vector. Clearly this is true in the case of $p$-adic numbers, since $F$ is the identity and $T$ is multiplication by $p$, but I'm struggling to prove the general case. Is the Frobenius map equivalent to the identity map on $W(k)$ in general?