  In the generalized truncation construction, one replaces each vertex of a
$k$-regular graph $\Gamma$ with a copy of a graph $\Upsilon$ of order $k$. We
investigate the symmetry properties of the graphs constructed in this way,
especially in connection to the symmetry properties of the graphs $\Gamma$ and
$\Upsilon$ used in the construction. We demonstrate the usefulness of our
results by using them to obtain a classification of cubic vertex-transitive
graphs of girths $3$, $4$, and $5$.
