A consequence of Itô's lemma

Question

In its simplest form, for any twice continuously differentiable function $f$ on the reals and Itô process $X_t$, it states that $f(t, X_t)$ is itself an Itô process. Every Itô process, for suitable conditions on drift and diffusion functions (see p.88 of "Stochastic Differential Equations and Applications" 2nd Edition, by X. Mao) is a Markov process. Does that mean $f(t, X_t)$ is Markov without further requirements on $f$? I read in another post that $f$ should be injective for this to be true, so twice differentiability doesn't seem enough. But then Itô's lemma wouldn't hold. I don't know