  A subgroup of the automorphism group of a graph $\G$ is said to be {\em
half-arc-transitive} on $\G$ if its action on $\G$ is transitive on the vertex
set of $\G$ and on the edge set of $\G$ but not on the arc set of $\G$.
Tetravalent graphs of girths $3$ and $4$ admitting a half-arc-transitive group
of automorphisms have previously been characterized. In this paper we study the
examples of girth $5$. We show that, with two exceptions, all such graphs only
have directed $5$-cycles with respect to the corresponding induced orientation
of the edges. Moreover, we analyze the examples with directed $5$-cycles, study
some of their graph theoretic properties and prove that the $5$-cycles of such
graphs are always consistent cycles for the given half-arc-transitive group. We
also provide infinite families of examples, classify the tetravalent graphs of
girth $5$ admitting a half-arc-transitive group of automorphisms relative to
which they are tightly-attached and classify the tetravalent
half-arc-transitive weak metacirculants of girth $5$.
