I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad_l h)\triangleright k=\sum h_1 kS(h_2)$ of a hopf algebra $H$ on itself.
(It is known that the left adjoint action of $H$ on itself is always inner (in the sense of the def. 6.1.1, p. 87, in Montgomery's book "Hopf algebras and their action on rings") and turns $H$ into a $H$-module algebra).
If you set $H=kG$, then, the left adjoint action for $g\in G$ reduces to a conjugation: $(ad_l g)\triangleright k=gkg^{-1}$.
More generally, for any inner action $\ \triangleright : H\otimes A\rightarrow A$, it is easy to show that any grouplike element $g\in G(H)$, acts as an inner automorphism of $A$ (see ch.6, example 6.1.5 in the same book for more details).
Concluding, i would say that the notion of an "inner action" (as described in the above reference) may be the reasonable notion you are looking for.