I have found it necessary to define the following property:

Consider a finite set $X$ and a group $H$ of permutations of $X$. Suppose for every normal subgroup $K$ of $H$ that $K$ acts faithfully on every $K$-orbit.

The property I have is that $G$ acts on $X$, not necessarily faithfully, inducing a permutation group $H$ as above.

Is there a standard name for the property possessed by $H$?

Certainly primitive permutation groups are examples for $H$, but so are fixed-point-free permutation groups for instance. On the other hand, $S_n \wr S_k$ acting on $nk$ points in the usual imprimitive way is not an example.

I think 'quasi-primitive' is already taken, and means 'every normal subgroup is transitive'. This is strictly stronger than the property I am talking about.