This question concerns locally compact groups equipped with Haar measure, $(G,\lambda)$. For a class of such groups, there exists an approximate identity $F_\nu$ such that the map $f\in L^1(G)\mapsto f\ast F_\nu\in L^1(G)$ is of finite rank for each $\nu$. A well known example is given by the Fejér kernels on the torus. Also for a certain class of compact groups which include finite dimensional unitary groups and its closed subgroups, there exist also such approximate identities: see for example Theorem 44.25, Hewitt & Ross, Abstract Harmonic Analysis, 2nd ed.)
I'm interested in knowing which other locally compact groups admit such approximate identities satisfying this additional finite rankness property. that is:
For which locally compact groups there exists an approximate identity $F_n$ such that the map $f\in L^1(G)\mapsto f\ast F_\nu\in L^1(G)$ is of finite rank for each $\nu$?
A more specific subquestion is:
Are there such approximate identities for all compact abelian groups?
NOTE: there is an answer here about the presence of approximate identities in $L^1(G)$, but in that thread the matter of finite rankness is not addressed.