  We study the problem #IndSub(P) of counting all induced subgraphs of size k
in a graph G that satisfy the property P. This problem was introduced by Jerrum
and Meeks and shown to be #W[1]-hard when parameterized by k for some families
of properties P including, among others, connectivity [JCSS 15] and even- or
oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two
of the authors introduced a novel technique for the complexity analysis of
#IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks
and Sturtevant [FOCS 83] and the framework of graph motif parameters due to
Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide
range of properties P. In this work, we refine this technique for graph
properties that are non-trivial on edge-transitive graphs with a prime power
number of edges. In particular, we fully classify the case of monotone
bipartite graph properties: It is shown that, given any graph property P that
is closed under the removal of vertices and edges, and that is non-trivial for
bipartite graphs, the problem #IndSub(P) is #W[1]-hard and cannot be solved in
time f(k)*n^{o(k)} for any computable function f, unless the Exponential Time
Hypothesis fails. This holds true even if the input graph is restricted to be
bipartite and counting is done modulo a fixed prime. A similar result is shown
for properties that are closed under the removal of edges only.
