Multiply set-transitive group action

Symbol-free definition
A group action on a set is termed $$k$$-set-transitive or $$k$$-homogeneous for $$k \ge 1$$ if the following is true for all $$1 \le l \le k$$:

Consider the natural action of the group on the set of subsets of size $$l$$. This action is a transitive group action.

A group action is termed multiply set-transitive or multiply homogeneous if it is $$k$$-set-transitive for some $$k > 1$$.

If a group action is $$k$$-set-transitive but not $$(k+1)$$-set-transitive, it is said to be sharply $$k$$-set-transitive.

Facts

 * The symmetric group on any set is $$k$$-set-transitive on it for every $$k$$.
 * The alternating group on any finite set (or more generally, the finitary alternating group on any set) is $$k$$-set-transitive on it for every $$k$$.

Stronger properties

 * Weaker than::Highly transitive group action
 * Weaker than::Highly set-transitive group action