The answer is yes for finite groups -- you can even ensure your space is a hyperbolic surface, or hyperbolic 3-manifold.  The result for 3-manifolds is Sadayoshi Kojima's.  For hyperbolic surfaces I forget who its due to, but I think the reference is in Kojima's paper. 

For compact Lie groups I think you can just make your space some variant of the Lie group itself (with a left-invariant metric).  The idea would be to take the Lie group $G$ with its left-invariant metric.  If that has a bigger isometry group than $G$ itself, take the product of $G$ with a ball (with some metric), and perturb the metric on $G \times B$, $G$-equivariantly.  Some generic perturbation of the metric should kill all isometries other than the ones coming from the $G$-action.