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Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$be partitions of $n$, where $\lambda=(\lambda_1,...,\lambda_n)$ and$\mu=(\mu_1,..,\mu_n)$. By $S^{\lambda}$ we denote the Specht modulecorresponding to $\lambda$ for the group algebra $K\mathfrak{S}_n$ of thesymmetric group $\mathfrak{S}_n$. D. Hemmer has raised the question of relatingthe homomorphism spaces $\Hom_{\mathfrak{S}_n}(S^{\mu}, S^{\lambda})$ and$\Hom_{\mathfrak{S}_{n'}}(S^{\mu^+}, S^{\lambda^+})$, where $n'=n+kp^d$,$\lambda^+ =\lambda+(kp^{d})$, $\mu^+=\mu+(kp^{d})$, and $d, k$ are positiveintegers. We show that these are isomorphic if $p$ is odd, $p^d>\min\{\lambda_2, \mu_1-\lambda_1\}$ and $\mu_2 \le \lambda_1$.

Relating homomorphism spaces between Specht modules of different degrees