  We prove that, given a finite graph $\Sigma$ satisfying some mild conditions,
there exist infinitely many tetravalent half-arc-transitive normal covers of
$\Sigma$. Applying this result, we establish the existence of infinite families
of finite tetravalent half-arc-transitive graphs with certain vertex
stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite
connected tetravalent half-arc-transitive graphs. This sheds some new light on
the longstanding problem of classifying the vertex stabilizers of finite
tetravalent half-arc-transitive graphs.
