A bicirculant is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we consider automorphisms of the so-called Taba\v cjn graphs, a family of pentavalent bicirculants which are obtained from the generalized Petersen graphs by adding two additional perfect matchings between the two orbits of the above mentioned automorphism. As a corollary, we determine which Taba\v cjn graphs are vertex-transitive.