  What is the true order of growth of torsion in the cohomology of an
arithmetic group? Let $D$ be a quaternion over an imaginary quadratic field
$F.$ Let $E/F$ be a cyclic Galois extension with $\mathrm{Gal}(E/F) = \langle
\sigma \rangle.$ We prove lower bounds for "the Lefschetz number of $\sigma$
acting on torsion cohomology" of certain Galois-stable arithmetic subgroups of
$D_E^\times.$ For these same subgroups, we unconditionally prove a
would-be-numerical consequence of the existence of a hypothetical base change
map for torsion cohomology.
