Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.

In other words, if in a transitive permutation group, each element that fixes a point, fixes exactly one point, and there is at least one such element we call it Frobenius.

Let us define a $t$-Frobenius group to be a transitive permutation group, such that each element that fixes a point fixes exactly $t$-points, and there is at least one such element.

Has this concept been studied already? Perhaps under different name?

Does $t$-Frobenius group exist for every $t$?

Does every $t$-Frobenius group have a regular subgroup?

Frobenius algebraic groupdue to David Hertzig in Amer. J. Math. 83 (1961), 421–431, as well as work on abstract infinite groups which generalize Frobenius groups. – Jim Humphreys Aug 6 '14 at 15:13