  We introduce the concept of {\it pseudocover}, which is a counterpart of {\it
cover}, for symmetric graphs. The only known example of pseudocovers of
symmetric graphs so far was given by Praeger, Zhou and the first-named author a
decade ago, which seems technical and hard to extend to obtain more examples.
In this paper, we present a criterion for a symmetric extender of a symmetric
graph to be a pseudocover, and then apply it to produce various examples of
pseudocovers, including (1) with a single exception, each Praeger-Xu's graph is
a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph
with vertex stabilizer of size divisible by $32$ has connected pseudocovers.
