  Given a finite-dimensional real vector space $V$, a probability measure $\mu$
on $\operatorname{PGL}(V)$ and a $\mu$-invariant subspace $W$, under a
block-Lyapunov contraction assumption, we prove existence and uniqueness of
lifts to $P(V)\setminus P(W)$ of stationary probability measures on the
quotient $P(V/W)$. In the other direction, i.e. under block-Lyapunov expansion,
we prove that stationary measures on $P(V/W)$ have lifts if any only if the
group generated by the support of $\mu$ stabilizes a subspace $W'$ not
contained in $W$ and exhibiting a faster growth than on $W \cap W'$. These
refine the description of stationary probability measures on projective spaces
as given by Furstenberg, Kifer and Hennion, and under the same assumptions,
extend corresponding results by Aoun, Benoist, Bru\`{e}re, Guivarc'h, and
others.
