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?