We are given that presentation of $S_4$ is $ \langle x,y|x^4=y^2=(x*y)^3=1 \rangle$ and $|S_4| = 24$.
My approach:
I have proven that $|\text{Aut}(Q_8)|=24$ and there exists $f$ and $g$ in $\text{Aut}(Q_8)$ such that $f^4 = \text{Id}$ and $g^2= \text{Id}$ and $(f*g)^3= \text{Id}$ (Where $f(i)=i$,$f(j)=k$ and $g(i)=j,g(j)=i$). If I prove that $f$ and $g$ generate the whole of $\text{Aut}(Q_8)$ then I am done, but this seems rather a tough task. I have speculations that after finding $f$ and $g$ is enough to prove it but I am not sure why.
It would be great if anyone could help.
Edit:
There is a similar question here Automorphism group of the quaternion group
But it doesn't prove by presentation of groups and I want the question to be solved in the way I have approached.