Let $G$ be a locally compact group. We know that if $G$ is abelian and $\cal F$ implies the Fourier transform, for every Segal algebra of $G$ say $S^1(G)$, ${\cal F}S^1(G)$ is an abstract Segal algebra with respect to $A(\widehat{G})$ (since $S^1(G)$ is an abstract Segal algebra of $L^1(G)$). Reiter in [1, Proposition 6.2.5] has shown that $C_c(\widehat{G})\cap A(\widehat{G}) \subseteq {\cal F}S^1(G)$.


My Question is:
Can we show that for an aribtarary locally compact group $G$ (not necessarily abelian) every abstract Segal algebra of $A(G)$ contains all functions in $A(G)$ that have compact support?


[1] H. Reiter, and J. D. Stegeman, Classical harmonic analysis and locally compact groups,
2nd edn, London Mathematical Society Monographs, New series 22, Oxford university press,
New York, 2000.