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In [Bryant, R. M.; Kovács, L. G., Lie representations and groups of prime power order. J. London Math. Soc. (2) 17 (1978), no. 415-421], the authors come close to proving what you ask: they show that any group $G < GL(V)$, with $V$ an elementary abelian $p$-group, can be realized as the image of $Aut(P) \rightarrow Aut(P/\Phi(P)) = GL(V)$, for some finite $p$--group $P$ with Frattini quotient $P/\Phi(P) = V$.

I used this in a paper with Frank Adams in the late 1980's, and we also reference Theorem 13.5 of Huppert and Blackburn, Finite Groups II. Section 13 of that book is all about automorphisms of $p$--groups. My memory (30 years old) is that exactly the result you are interested in was proved there. (This is not available online, so I can't easily check this right now.)

Added a little bit later: I should perhaps have explicitly pointed out that the map $Aut(P) \rightarrow Aut(P/\Phi(P))$ factors through $Aut(P) \rightarrow Out(P)$. Also the kernel of this map is a $p$--group: see e.g. [ (24.1), Aschbacher, Finite Group Theory ]. So one knows that, for any prime $p$, any finite group is the quotient of $Out(P)$ for some finite $p$--group $P$, with another $p$--group as kernel.

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}$In [Bryant, R. M.; Kovács, L. G., Lie representations and groups of prime power order. J. London Math. Soc. (2) 17 (1978), no. 415-421], the authors come close to proving what you ask: they show that any group $G < \GL(V)$, with $V$ an elementary abelian $p$-group, can be realized as the image of $\Aut(P) \rightarrow \Aut(P/\Phi(P)) = \GL(V)$, for some finite $p$-group $P$ with Frattini quotient $P/\Phi(P) = V$.

I used this in a paper with Frank Adams in the late 1980s, and we also reference Theorem 13.5 of Huppert and Blackburn, Finite Groups II. Section 13 of that book is all about automorphisms of $p$-groups. My memory (30 years old) is that exactly the result you are interested in was proved there. (This is not freely available online, so I can't easily check this right now.)

Added a little bit later: I should perhaps have explicitly pointed out that the map $\Aut(P) \rightarrow \Aut(P/\Phi(P))$ factors through $\Aut(P) \rightarrow \Out(P)$. Also the kernel of this map is a $p$-group: see e.g. [ (24.1), Aschbacher, Finite Group Theory ]. So one knows that, for any prime $p$, any finite group is the quotient of $\Out(P)$ for some finite $p$-group $P$, with another $p$-group as kernel.

In [Bryant, R. M.; Kovács, L. G., Lie representations and groups of prime power order. J. London Math. Soc. (2) 17 (1978), no. 415-421], the authors come close to proving what you ask: they show that any group $G < GL(V)$, with $V$ an elementary abelian $p$-group, can be realized as the image of $Aut(P) \rightarrow Aut(P/\Phi(P)) = GL(V)$, for some finite $p$--group $P$ with Frattini quotient $P/\Phi(P) = V$.

I used this in a paper with Frank Adams in the late 1980's, and we also reference Theorem 13.5 of Huppert and Blackburn, Finite Groups II. Section 13 of that book is all about automorphisms of $p$--groups. My memory (30 years old) is that exactly the result you are interested in was proved there. (This is not available online, so I can't easily check this right now.)

Added a little bit later: I should perhaps have explicitly pointed out that the map $Aut(P) \rightarrow Aut(P/\Phi(P))$ factors through $Aut(P) \rightarrow Out(P)$, so one knows that any finite group is the quotient of $Out(P)$ for some finite $p$--group $P$, for any prime $p$.

In [Bryant, R. M.; Kovács, L. G., Lie representations and groups of prime power order. J. London Math. Soc. (2) 17 (1978), no. 415-421], the authors come close to proving what you ask: they show that any group $G < GL(V)$, with $V$ an elementary abelian $p$-group, can be realized as the image of $Aut(P) \rightarrow Aut(P/\Phi(P)) = GL(V)$, for some finite $p$--group $P$ with Frattini quotient $P/\Phi(P) = V$.

I used this in a paper with Frank Adams in the late 1980's, and we also reference Theorem 13.5 of Huppert and Blackburn, Finite Groups II. Section 13 of that book is all about automorphisms of $p$--groups. My memory (30 years old) is that exactly the result you are interested in was proved there. (This is not available online, so I can't easily check this right now.)

Added a little bit later: I should perhaps have explicitly pointed out that the map $Aut(P) \rightarrow Aut(P/\Phi(P))$ factors through $Aut(P) \rightarrow Out(P)$. Also the kernel of this map is a $p$--group: see e.g. [ (24.1), Aschbacher, Finite Group Theory ]. So one knows that, for any prime $p$, any finite group is the quotient of $Out(P)$ for some finite $p$--group $P$, with another $p$--group as kernel.

In [Bryant, R. M.; Kovács, L. G., Lie representations and groups of prime power order. J. London Math. Soc. (2) 17 (1978), no. 415-421], the authors come close to proving what you ask: they show that any group $G < GL(V)$, with $V$ elementary abelian, can be realized as the image of $Aut(P) \rightarrow Aut(P/\Phi(P)) = GL(V)$, for some finite $p$--group $P$ with Frattini quotient $P/\Phi(P) = V$.

I used this in a paper with Frank Adams in the late 1980's, and we also reference Theorem 13.5 of Huppert and Blackburn, Finite Groups II. Section 13 of that book is all about automorphisms of $p$--groups. My memory (30 years old) is that exactly the result you are interested in was proved there. (This is not available online, so I can't easily check this right now.)

Added a little bit later: I should perhaps have explicitly pointed out that the map $Aut(P) \rightarrow Aut(P/\Phi(P))$ factors through $Aut(P) \rightarrow Out(P)$, so one knows that any finite group is the quotient of $Out(P)$ for some finite $p$--group $P$, for any prime $p$.

In [Bryant, R. M.; Kovács, L. G., Lie representations and groups of prime power order. J. London Math. Soc. (2) 17 (1978), no. 415-421], the authors come close to proving what you ask: they show that any group $G < GL(V)$, with $V$ an elementary abelian $p$-group, can be realized as the image of $Aut(P) \rightarrow Aut(P/\Phi(P)) = GL(V)$, for some finite $p$--group $P$ with Frattini quotient $P/\Phi(P) = V$.

I used this in a paper with Frank Adams in the late 1980's, and we also reference Theorem 13.5 of Huppert and Blackburn, Finite Groups II. Section 13 of that book is all about automorphisms of $p$--groups. My memory (30 years old) is that exactly the result you are interested in was proved there. (This is not available online, so I can't easily check this right now.)

Added a little bit later: I should perhaps have explicitly pointed out that the map $Aut(P) \rightarrow Aut(P/\Phi(P))$ factors through $Aut(P) \rightarrow Out(P)$, so one knows that any finite group is the quotient of $Out(P)$ for some finite $p$--group $P$, for any prime $p$.