  We show that a generically sharply $t$-transitive permutation group of finite
Morley rank on a set of rank $r$ satisfies $t\le r+2$ provided the pointwise
stabilizer of a generic $(t-1)$-tuple is an $L$-group, which holds, for
example, when the stabilizer is solvable or when $r\le 5$. This makes progress
on the Borovik-Cherlin conjecture that every generically $(r+2)$-transitive
permutation group of finite Morley rank on a set of rank $r$ is of the form
$\operatorname{PGL}_{r+1}(F)$ acting naturally on $\mathbb{P}^r(F)$.
  Our proof is assembled from three key ingredients that are independent of the
main theorem -- these address actions of $\operatorname{Alt}(n)$ on $L$-groups
of finite Morley rank, generically $2$-transitive actions with abelian point
stabilizers, and simple groups of rank $6$.
