  Let $X$ be a connected scheme, smooth and separated over an algebraically
closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a
smooth proper morphism and $x$ a geometric point on $X$. We prove that the
tensor invariants of bounded length $\leq d$ of $\pi_1(X,x)$ acting on the
\'etale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of
those of $\pi_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a
constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to
show that the geometric variant with $F_\ell$-coefficients of the
Grothendieck-Serre semisimplicity conjecture -- namely that $\pi_1(X,x)$ acts
semisimply on $H^*(Y_x,F_\ell)$ for $\ell\gg 0$ -- is equivalent to the
condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,Q_\ell)$ is `almost
maximal' (in a precise sense; what we call `almost hyperspecial') with respect
to the group of $Q_\ell$-points of its Zariski closure. Ultimately, we prove
the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre
semisimplicity conjecture.
