  This paper is a significant contribution to a general programme aimed to
classify all projective irreducible representations of finite simple groups
over an algebraically closed field, in which the image of at least one element
is represented by an almost cyclic matrix (that is, a square matrix $M$ of size
$n$ over a field $F$ with the property that there exists $\alpha\in F$ such
that $M$ is similar to $diag(\alpha \cdot Id_k, M_1)$, where $M_1$ is cyclic
and $0\leq k\leq n$). While a previous paper dealt with the Weil
representations of finite classical groups, which play a key role in the
general picture, the present paper provides a conclusive answer for all
cross-characteristic projective irreducible representations of the finite
quasi-simple groups of Lie type and their automorphism groups.
