  The century old extremal problem, solved by Carath\'eodory and Fej\'er,
concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the
quantity to be maximized is the coefficient a1. In the complex exponential
form, the coefficient sequence (ck) will be supported in [-n; n] and normalized
by c0 =1. Reformulating, nonnegativity of T translates to positive definiteness
of the sequence (ck), and the extremal problem becomes a maximization problem
for the value at 1 of a normalized positive definite function c : Z --> C,
supported in [-n; n]. Boas and Katz, Arestov, Berdysheva and Berens,
Kolountzakis and R\'ev\'esz and recently Krenedits and R\'ev\'esz investigated
the problem in increasing generality, reaching analogous results for all
locally compact Abelian groups. We prove an extension to all the known results
in not necessarily commutative locally compact groups.
