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Couple of days ago, i actually asked this question : Existence of a normal subgroup with $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$.

I was thinking about the converse of this statement. Suppose $G$ is a finite group, with subgroups $H$ and satisfies $|Aut(G)|=|Aut(H)|,$ then can anything be said about, $H$ or $G$ in terms of their structure. Will $H$ be normal or ....? Or one can even ask, that find all finite groups $G$ such that $|Aut(H)|=|Aut(G)|$ for every subgroup $H$ of $G$! I am not interested that much in this question as there seems one can have lot of Groups of this type, but i am more curious to know the behaviour of $H$.

Also, anyone interested can very well read this article in MO: https://mathoverflow.net/questions/1075/order-of-an-automorphism-of-a-finite-group

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    @Steve D: Ok steve, thanks. If supposing i invoke, some condition do you think that it can be solved. For example $H$ is abelian....something like that2010-08-18

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In response to:

...find all finite groups $G$ such that $|\mathrm{Aut}(H)|=|\mathrm{Aut}(G)|$ for every subgroup $H$ of $G$

the only groups (finite or infinite) for which this is true is the trivial group and the group of order $2$.

Proof: All groups have the trivial group $\{1\}$ as a subgroup, which has a trivial automorphism group, i.e., $|\mathrm{Aut}(\{1\})|=1$. By the assumptions on $G$, we have $|\mathrm{Aut}(\{1\})|=|\mathrm{Aut}(G)|=1$. The only groups with $|\mathrm{Aut}(G)|=1$ are the trivial group and the group of order $2$ (ref.). The property is true for these two groups.

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    Even without including the trivial subgroup, any cyclic subgroup of order $p$ has automorphism group of size $p-1$. So if $|G|$ is divisible by two primes, you're out of luck. If $G$ is a p-group, looking at $G/\Phi(G)$ shows it would have to be cyclic. A quick count then shows $G$ would have to be cyclic of prime order.2013-05-01