  Let $X$ be a minimal projective threefold of general type over $\mathbb{C}$
with only Gorenstein quotient singularities, and let
$\mathrm{Aut}_{\mathbb{Q}}(X)$ be the subgroup of automorphisms acting
trivially on $H^*(X,\mathbb{Q})$. In this paper, we show that if $X$ is of
maximal Albanese dimension, then $|\mathrm{Aut}_{\mathbb{Q}}(X)|\leq 6$.
Moreover, if $X$ is nonsingular and $K_X$ is ample, then
$|\mathrm{Aut}_{\mathbb{Q}}(X)|\leq 5$.
  Seeking for higher-dimensional examples of varieties with nontrivial
$\mathrm{Aut}_{\mathbb{Q}}(X)$, we concern $d$-folds $X$ isogenous to an
unmixed product of curves. If $d=3$, we show that
$\mathrm{Aut}_{\mathbb{Q}}(X)$ is a $2$-elementray abelian group whose order is
at most $4$ under some conditions on their minimal realizations. Moreover, each
of the possible groups can be realized. If $d\geq 3$, we give a sufficient
condition for $\mathrm{Aut}_{\mathbb{Q}}(X)$ being trivial.
  Curiously, there exist examples of projective threefolds $X$ with terminal
singularities and maximal Albanese dimension whose
$\mathrm{Aut}_{\mathbb{Q}}(X)$ can have an arbitrarily large order.
