Let $G$ be a finite group and we take for the field the complex numbers. Call an irreducible character $\xi$ with degree $m$ of $G$ perfect, if all exterior powers $\bigwedge\nolimits^k \xi$ are irreducible for $k=1,\dotsc,m$.

Question 1: Can perfect irreducible characters be characterised in an easy way?

Question 2: What are the perfect irreducible characters for the symmetric group?

A famous example is the standard representation, but there are more. For example $S_6$ has 6 such irreducible perfect characters.

Question 3: What are the finite groups having a faithful irreducible character that is perfect? For $2$-groups the number of such groups seems rather small and starts for 2-groups with $n \geq 2$ elements with 1,1,3,6,9,12,15.