I am actually going through "Twenty Four Hours of Local Cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:

Find a ring $R$ of characteristic $p$ with the property that its Frobenius endomorphism $\varphi:R\rightarrow R$ is not flat but $R$ is flat over $\varphi(R)$.

The Frobenius endomorphism $\varphi:R\rightarrow R$ is the map $\varphi(r)=r^p$.

I am playing around with the ring $k[[x^2,xy,y^2]]$ for which I know the frobenius endomorphism is not flat, but somehow i am not able to prove that $R$ is flat over $\varphi(R)$.