When G is a non-amenable locally compact group, the full group C*-algebra C^*(G) and the reduced group C*-algebra C^*_r(G) differ. An exotic group C*-algebra of such a group is a C*-algebra that lies between'' C^*(G) and C^*_r(G). We will discuss a new construction of potentially exotic group C*-algebras arising the enveloping C*-algebras of PF^*_p(G) (1<= p <= infty) -- a newly introduced class of Banach *-algebras associated with G. As a byproduct of our techniques, we give a near solution to a conjecture of Cowling.