A group is called locally finite if every finitely generated subgroup is finite. The authors study some spectral properties of random walks that are symmetric and invariant under translations on infinite countable groups with an emphasis on locally finite groups. Among other results it is shown that any locally compact unimodular non-compact amenable group carries an irreducible symmetric random walk whose return probability decay is faster than any given sub-exponential function. The authors also construct, on any infinite countable locally finite group, random walks with return probabilities whose decay is slower than any given positive function which goes to zero as time goes to infinity. At the end of the paper the authors provide a table that summarizes the computed results, showing for a given measure \(\mu\) on a locally finite group the spectral profile, the spectral density and the return probability associated to the measure \(\mu\).