Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$.

Let $M$ be a finitely generated $kG$-module.

We denote the first syzygy of $M$ by $\Omega(M)$, i.e. $\Omega(M):=\text{Ker}(p)$ where $P\stackrel{p}{\rightarrow} M$ is a minimal projective cover of $M$.

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A $kG$-homomorphism $m: M\rightarrow \Omega(M)$ is called **almost projective**,

if $m$ is a generator of the simple socle of $\text{Hom}_{kG}(M,\Omega(M))/\text{Proj}_{kG}(M,\Omega(M))$ as $\text{End}_{kG}(M)$-module.

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Here, $\text{Proj}_{kG}(M,\Omega(M))$ is the set of all projective $kG$-homomorphisms from $M$ to $\Omega(M)$ (i.e. those that factor through a projective $kG$-module).

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I would like to ask the following question:

Can MAGMA produce / compute such an almost projective $kG$-homomorphism $m$?

Thank you very much for the help.

EDIT (9th April): The motivation is to construct almost split sequences with MAGMA and this is related to the following earlier question: