Letting $G$ be a group and $S(G)$ be all permutations of $G$, define
$$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$
It is easy to check that $L(G)$ is a group. My intuition to define $L(G)$ is to define "local isomorphisms " of a group as it sends cyclic groups to its correspondonce. We have $L(G)\geq Aut(G)$, and if $G$ is cyclic then $L(G)=Aut(G)$.
With similiar intuition, we can define
$$A(G)=\{\phi\in S(G)|\phi(xy)=\phi(x)\phi(y)\ \ if \ xy=yx \}$$
and can see that $L(G)\geq A(G)\geq Aut(G)$, and if $G$ is abelian then $A(G)=Aut(G)$.
The group $Aut(G)$ may give a lot of information about $G$. Is that also true about $L(G)$ or $A(G)$?
Any further observations would be appreciated.
I cannot tell whether this concept is useful or not. It may be a fun thing to play elementary games with, though.
Consider the case where $G$ is a $p$-group of exponent $p$. For example, an elementary $p$-abelian group or an extra special group. In those cases we can partition the non-identity elements of $G$ into $N=(|G|-1)/(p-1)$ subsets $G(i)$ such that $G(i)$ consists of the $p-1$ non-identity powers of an element $x_i, i=1,\ldots,N.$ The sets $G(i)$ are the equivalence classes of the relation on $G\setminus\{1\}$: $x\sim y$ iff $x=y^a$ for some integer $a$ coprime to $p$, and $D=\{x_i\mid i=1,2,\ldots,n\}$ is a set of representatives.
In this case we see that an element $\phi\in L(G)$ is fully determined by the images $\phi(x_i)$. Furthermore, the rule $$ \phi(x_i)=x_{\sigma(i)}^{a_i} $$ specifies a vector $\vec{a}:=(a_1,a_2,\ldots,a_N)\in (\Bbb{Z}_p^*)^N$ and a permutation $\sigma\in Sym(D)$ - both uniquely determined by $\phi$. Conversely any such vector $\vec{a}$ and a permutation $\sigma$ give rise to an element of $L(G)$.
Let's denote the above mapping $\phi$ by $\phi(\vec{a},\sigma)$. For a composition of two such mappings we have $$ \phi(\vec{a},\sigma)\circ\phi(\vec{b},\tau):x_i\mapsto x_{\tau(i)}^{b_i}\mapsto x_{\sigma\tau(i)}^{b_i a_{\tau(i)}}. $$ In other words $$ \phi(\vec{a},\sigma)\circ\phi(\vec{b},\tau)=\phi(\vec{a}^\tau*\vec{b},\sigma\tau). $$ This shows that the group $L(G)$ is isomorphic to the wreath product $$ L(G)\cong \Bbb{Z}_p^*\wr S_N. $$
This group is much larger than $Aut(G)$. For example when $G=\Bbb{F}_p^n$ is the elementary $p$-Abelian group, then $Aut(G)=GL_n(\Bbb{F}_p)$.
When all the non-identity elements of $G$ are of prime order a similar structure is there for we get one factor for each prime. When there are elements of composite order, then the fun begins, for we get extra constraints from (possibly shared) powers of elements of a given order.