A function is determined by the norms of its convolutions

Let $G$ be a compact abelian group and $f$, $g\in L^p(G)$, $p$ is not even. Let $\varphi_a$ denote the $a$-shift of the function $\varphi$. It is proved that
$$ \bigl\|\sum\alpha_if_{a_i}\bigr\|_{L^p}=\bigl\|\sum\alpha_ig_{a_i}\bigr\|_{L^p}, $$
for all $\alpha_1,\dots,\alpha_n\in\mathbb R^1$ and $a_1,\dots,a_n\in G$ then there exist $b\in G$ and $\alpha\in\mathbb R^1$, $|\alpha|=1$, such that $f=\alpha g_b$.