Mapping class between coverings of Riemann surfaces Let $X$ be a closed Riemann surfaces of genus $g$ and let $p_1:Y_1 \rightarrow X$ and $p_2:Y_2 \rightarrow X$ be two $K$-sheeted, connected, unramified coverings of the Riemann surface. By the theorem of Riemann-Hurwitz the genus of $Y_1$ and $Y_2$ is $K(g-1)+1$.
So there is an orientation preserving diffeomorphism between $Y_1$ and $Y_2$.
My question is if there is always a mapping class which maps $Y_1$ to $Y_2$. 
 A: This is a simple exercise and the answer is no (cf. comments of Lee Mosher and Misha). Consider the two degree-4 coverings of the torus to itself with monodromies  $\omega_1, \omega_2 : \pi_1(T^2) \to \Sigma_4$ given by $\omega_1(e_1) = (1\ 2\ 3\ 4)$, $\omega_1(e_2) = 1$ and $\omega_2(e_1) = (1\ 2) (3\ 4)$, $\omega_2(e_2) = (2\ 3) (1\ 4)$ (with respect to a basis $\{e_1, e_2\}$ for $\pi_1(T^2)$). These are in different orbits with respect to the action of $\text{Map}(T^2) \cong SL(2, \Bbb Z)$. As a further exercise prove that for coverings of the torus of degree $\leq 3$ the action is transitive.
A: Let $F$ be a finite group, let $\pi_g$ be the fundamental group of the closed oriented genus $g$ surface $S_g$ and let $Mod_g$ denote the mapping class group of $S_g$. Consider the action of $Mod_g$ on the "character variety'' $R(\pi_g,F)=Hom(\pi_g, F)/Aut(F)$ by precompositions. 
The question is "how transitive is the action of $Mod_g$ on $R(\pi_g,F)$?" It is clear (see e.g. Daniele's answer) that $Mod_g$ cannot send an epimorphism to a non-epimorphism. Thus, consider the subset $E(\pi_g,F)\subset R(\pi_g, F)$ consisting of equivalence classes of epimorphisms. There is one more invariant that $Mod_g$ has to preserve, namely, every homomorphism $f: \pi_g\to F$ represents an element $c_f$ of $H_2(F, {\mathbb Z})$, which is the image of the fundamental class of $S_g$ under the induced map 
$$H_2(f): H_2(S_g)\to H_2(K(F,1)).$$ 
Thus, we get a map $c: R(\pi_g,F)\to Q_F:=H_2(F, {\mathbb Z})/Aut(F), f\mapsto c_f$. The classes $c_f$ (modulo $Aut(F)$) are, clearly, preserved by the action of $Mod_g$, so $Mod_g$ preserves each fiber of the map $c$.  
Amazingly, it turns out that for every simple nonabelian group $Q$, "stably" (i.e., for fixed $F$ and all sufficiently  large genera $g$), the map $c$ completely classifies the $Mod_g$-orbits on $E(\pi_g,F)$: 
For every simple nonabelian $Q$, if $g$ is sufficiently large, two epimorphisms $f_1, f_2$ belong to the same orbit $\iff$ $c_{f_1}=c_{f_2}$. This is proven in Theorem 1.3 of the paper by N.Dunfield and W.Thurston "Finite covers of random 3-manifolds". Furthermore, for all large $g$, the action of $Mod_g$ on every orbit $O$ is via the full alternating group of $O$. What happens for small $g$'s is very interesting but unclear. 
A similar result, based on the previous work of Convey and Parker, in the context of the action of the braid group, was proven in: M. D. Fried and H. Volklein, "The inverse Galois problem and rational points on moduli spaces," Math. Ann. 290 (1991), 771–800, see footnote on page 5 of the paper by Dunfield and Thurston. 
