Stability automorphism of subnormal series

Symbol-free definition
An automorphism of a group is said to be a stability automorphism with respect to a subnormal series if it induces the identity map on each successive quotient for the subnormal series.

The stability automorphisms of any fixed subnormal series form a group, called the stability group of that subnormal series. This group lives as a subgroup of the automorphism group.

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is termed a stability automorphism with respect to the subnormal series:

$$\{ e \} = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_n = G$$

if $$\sigma(H_ix) = H_ix$$ for any $$x \in H_{i+1}$$, or equivalently, $$\sigma$$ acts as identity on $$H_{i+1}/H_i$$.

(An analogous definition can be given for subnormal series indexed by infinite sets).

Facts

 * Stability group of subnormal series of p-group is p-group