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I was reading This question on MO and suddenly faced another question.

Is there any necessary and sufficient condition for $G$ such that there exist a group $G'$ that $G \cong Aut(G')$ ? What can we say about subgroups of $G$ when we know the group $G'$ with that condition exist.

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    $\begingroup$ I would rather not call the second group $G'$, in order to avoid confusion with the common notation for the derived subgroup. $\endgroup$
    – Stefan Kohl ♦
    Commented May 26, 2014 at 10:27
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    $\begingroup$ Personally I avoid using $G'$ for the derived group for the same reason! :) $\endgroup$
    – Derek Holt
    Commented May 26, 2014 at 10:38
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    $\begingroup$ There is some literature on this topic. For example, if $n > 1$ is odd and has no divisor $p^4$ for any prime $p$, then there is no finite group $G$ such that $|{\rm Aut}(G)| = n$: link.springer.com/article/10.1007/BF02884017 $\endgroup$
    – Derek Holt
    Commented May 26, 2014 at 10:46
  • $\begingroup$ You should look at this paper: projecteuclid.org/euclid.rmjm/1250129106 In particular, note that there are at most a finite number of $G'$ satisfying your equation. $\endgroup$
    – Nick Gill
    Commented May 26, 2014 at 18:01

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Let me expand on my comment. There are two papers that you should look at:

Iyer, Hariharan K. On solving the equation $Aut(X)=G$. Rocky Mountain J. Math. 9 (1979), no. 4, 653–670.

The scope of this paper is when your group $G'$ is finite. The MathSciNet review gives a perfect precis:

The author considers the finite groups having a given automorphism group $H$. First he shows (Theorem 3.1) that for every finite group H there are only finitely many finite groups $G$ having $H$ as automorphism group, and he surveys the possible groups $G$ in the cases when $H$ is one of the following types of group: alternating, symmetric, dihedral, dicyclic, quasidihedral (Theorems 4.2, 4.4, 6.3 and 6.6).

The second paper you should consult does not restrict $G'$ to being finite:

Robinson, Derek J. S. Groups with prescribed automorphism group. Proc. Edinburgh Math. Soc. (2) 25 (1982), no. 3, 217–227.

Again the MSN review tells the story:

The author treats the question of whether a given finite group is the full automorphism group of at least one group. The main theorem gives a description of all groups $G$ such that $Aut(G)$ is finite and $Aut_c(G)$, the subgroup of central automorphisms, is semisimple. Using this, he shows that for a finite simple (nonabelian) group $S$ there is a group $G$ such that $Aut(G)≅S$ if and only if one of the following is true: (a) $S=PSL(r,2)$, $r>2$, and $G$ is elementary abelian, (b) $S$ is complete, and $G\cong \hat{S}/K$ or $(\hat{S}/K)\times Z_2$, where $K$ is any subgroup of $M(S)$, the Schur multiplier of $S$, and $|M(S):K|$ is odd in the second case, (c) $S=Sz(8)$ and $G\cong\hat{S}/K$, $K<M(S)$, $|K|=2$..... For a finite group $S$ there are only finitely many finite groups G with $Aut(G)\cong S$. This is false if $G$ is infinite. For $S\cong S_4$ the author constructs uncountably many infinite nonabelian groups G with $Aut(G)\cong S_4$.

Note that the result detailing which simple groups are automorphism groups of an arbitrary group suggests that a general necessary and sufficient condition seems unlikely to exist.

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    $\begingroup$ All countable groups by Bumagin and Wise are outer automorphism groups so truly there is no hope. $\endgroup$
    – Benjamin Steinberg
    Commented May 26, 2014 at 21:39

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