intersection of conjugate subgroups

Let G be a group, H a subgroup of index k.
Denote by H0 the intersection of all conjugate subgroups of H.

Why does the index of H0 in G divide k! (factorial) ?

Dividing out H0, the question reduces to proving that if a g in G fixes
all conjugates of H, then g=1. But I don't know how to continue from
here.
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