The thesis deals with the question of (non-)invertibility of $C^*$-extensions arising from group $C^*$-algebras. The first chapter outlines the basic preliminaries and sets the stage. In particualar the basics of $C^*$-extensions and group $C^*$-algebras are treated and more advanced theory such as semi-invertibility of extensions and weak containment of representations is explained.

In the second chapter we give a negative result showing that the reduced group $C^*$-algebra of an amalgamated free product of Abelian groups has non-invertible extensions by the compact operators.

The third chapter gives a positive result. We show that all extensions of the reduced group $C^*$-algebra of a free product of amenable groups by any stable and sigma-unital $C^*$-algebra are semi-invertible.