The stack BG only recovers G up to inner automorphisms, not canonically (as suggested by blah) - this can lead to serious issues in families or equivalently over a nonalgebraically closed field, as Shenghao's comment points out. One way to say this is the following  : the loops in BG are G/G, the adjoint quotient of G. On the other hand, if you give a map pt --> G then the based loop space (fiber product of pt with itself over BG) is G, so you recover the group canonically.

The stack $BG$ only recovers $G$ up to inner automorphisms, not canonically (as suggested by blah) - this can lead to serious issues in families or equivalently over a nonalgebraically closed field, as Shenghao's comment points out. One way to say this is the following: the loops in $BG$ are $G/G$, the adjoint quotient of $G$. On the other hand, if you give a map $pt \to BG$ then the based loop space (fiber product of $pt$ with itself over $BG$) is $G$, so you recover the group canonically.