Let H be a group. Prove there is a group G containing a normal subgroup H' with the following two properties:

(a) The group H is isomorphic to H' , and

(b) For every automorphism $\displaystyle \phi$ of H', there is an element $\displaystyle g\inG$ such that

$\displaystyle \phi = I_{g}$ on H', where $\displaystyle I_{g}$ is the inner automorphism associated to g.

Thanks.