Action via automorphism

I want to ask what does it mean to say a group $A$ acts on $N$ via automorphisms. It is a notion used in M.Isaacs book and I am not familiar with. I tried to find how it is defined but a scanned e book is so hard to go through.

Suppose $*: A\times N \to N$ is a group action of $A$ on $N$, thus we have homomorphism $\phi: A \to S_N$. Now what does action via automorphism mean. Thanks!

Isaacs defines this notion on page 68 of his finite group theory here. To keep this answer available, let me restate it:

Given groups $\mathcal A = (A,\cdot_A, 1_A), \mathcal B = (B, \cdot_B, 1_B)$ we say that $\mathcal A$ acts on $\mathcal B$ via automorphisms iff there is a (left-)group action $\ast \colon \mathcal A \times \mathcal B \to \mathcal B$ such that for all $a \in A$ and all $b,c \in B$ we have $a \ast (b \cdot_B c) = (a \ast b) \cdot_B (a \ast c)$.