  A vertex-transitive map $X$ is a map on a closed surface on which the
automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If
the face-cycles at all the vertices in a map are of two types, then the map is
said to be a $2$-semiequivelar map. A 2-uniform tiling is an edge-to-edge
tiling of regular polygons having $2$ distinct transitivity classes of
vertices. Clearly, a $2$-uniform map is $2$-semiequivelar. The converse of this
is not true in general. We know that there are infinitely many tilings of
$2$-semiequivelar type. In this article, we study $2$-semiequivelar maps on the
torus that are the quotient of the plane's $2$-unform latices. We have found
bounds of the number of orbits of vertices of the quotient $2$-semiequivelar
maps on the torus under the automorphism groups (symmetric groups).
