Extending Semilinear Transformations over Finite Fields

Question

Suppose $ p $ a prime integer and $ m $ and $ n $ positive integers, where $ m | n $. Let $ \Phi_{m} $ denote the Frobenius automorphism of $ \mathbb{F}_{p^m} $ and $ \Phi_{n} $ the Frobenius automorphism of $ \mathbb{F}_{p^n} $. Given a $ \Phi_{m} $-semilinear transformation $ T \colon V \to W $ of $ \mathbb{F}_{p^m} $-vector spaces, is there a (known) way of extending $ T $ to a $ \Phi_{n} $-semilinear transformation $ V \otimes_{\mathbb{F}_{p^m}} \mathbb{F}_{p^n} \to W \otimes_{\mathbb{F}_{p^m}} \mathbb{F}_{p^n} $? (I.e., extension of scalars for semilinear transformations.)