We define another invariant for a finite group $G$, the so-called {\it covariant dimension of $G$} which is closely related to the essential dimension of $G$ which we introduced in the previous talk. It is the minimal possible dimension of the image of a faithful (regular) covariant of $G$. It turns out that many results for the essential dimension of a finite group $G$ can be carried over to the covariant dimension of $G$ where they admit surprisingly simple proofs. In this way we have been able to reconstruct the main results of Buhler and Reichstein for the essential dimension and their applications to field extensions. But still there remain a number of interesting open questions which are quite elementary and easy to understand. Some of them will be discussed at the end of the talk.