Yes, ${\rm PGL}(2,q^2)$ is a subgroup of ${\rm PGL}(4,q)$, but I would guess that that is an exception, and in general there is no such embedding.
${\rm GL}(4,q)$ contains the subgroup that I denote by ${\rm CO}^-(4,q)$, which is the conformal orthogonal group of minus-type (and equal to the normalizer in ${\rm GL}(4,q)$ of ${\rm GO}^-(4,q)$).
The projective image ${\rm PCO}^-(4,q)$, which is of course a subgroup of ${\rm PGL}(4,q)$, happens to be isomorphic to ${\rm P \Gamma L}(2,q^2)$, which contains ${\rm PGL}(2,q^2)$ as a subgroup of index $2$.
So this is a separate embedding, and is not related to the semiliear embedding ${\rm GL}(2,q^2) \to {\rm GL}(4,q)$.
The general embedding ${\rm PGL}(n,q^h) \to {\rm PGL}(n^h,q)$ can be defined as follows. Let $M$ be the natural module for $G={\rm GL}(n,q^h)$ and let $\phi$ be the field automorphism of $G$ induced by the automorphism $x \mapsto x^q$ of the field. Then $M \otimes M^\phi \otimes M^{\phi^2} \otimes \cdots \otimes M^{\phi^{h-1}}$ is a module of dimension $n^h$ that is stabilized by $\phi$ and hence can be realized over ${\mathbb F}_q$. So we get a homomorphism ${\rm GL}(n,q^h) \to {\rm GL}(n^h,q)$, which is not always injective, but scalars map to scalars, so it induces the required embedding ${\rm PGL}(n,q^h) \to {\rm PGL}(n^h,q)$.