I have tried to solve it in the following way though I find little bit difficulty to reach at the desired conclusion.Here's my way.
I take the action of $A_5$ on the set of all left cosets of $H$ in $G$ by left multiplication where $H$ is the subgroup of order $20$ in $A_5$.Then this action induces a homomorphism from $A_5$ into $S_3$.If we observe the kernel of that homomorphism then we find that it either contains only the identity or it is whole of $A_5$ by the simplicity of $A_5$.But it cannot contain only the identity for which case we have a monomorphism from $A_5$ into $S_3$ which cannot be possible since $|A_5| > |S_3|$. So it is the whole of $A_5$.But here I got stuck.It may happen in which case the homomorphism becomes trivial.But to arrive at a contradiction I must have some argument which I have failed to crack in my mind.
Would anybody please help me in this regard for finding some way out.Then that will really help me a lot.
Thank you in advance.