  A graph is edge-transitive if the natural action of its automorphism group on
its edge set is transitive. An automorphism of a graph is semiregular if all of
the orbits of the subgroup generated by this automorphism have the same length.
While the tetravalent edge-transitive graphs admitting a semiregular
automorphism with only one orbit are easy to determine, those that admit a
semiregular automorphism with two orbits took a considerable effort and were
finally classified in 2012. Of the several possible different ``types'' of
potential tetravalent edge-transitive graphs admitting a semiregular
automorphism with three orbits, only one ``type'' has thus far received no
attention. In this paper we focus on this class of graphs, which we call the
Woolly Hat graphs. We prove that there are in fact no edge-transitive Woolly
Hat graphs and classify the vertex-transitive ones.
