Let $G$ be a finite group of order $240$. If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are $ [1,1,1,1,~3,3,3,3,3,3,3,3, ~4,4,4,4,~5,5,5,5 ]. $

Conversely, Suppose that $G$ is non-solvable, and the all degrees of irreducible $\mathbb{C}$-characters are $ [1,1,1,1,~3,3,3,3,3,3,3,3, ~4,4,4,4,~5,5,5,5 ]. $

Question: $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?