  We prove that for any Hecke eigenform f of level 1 and arbitrary weight there
is a self-dual cuspidal automorphic form $\pi$ of $GL_6(\Q)$ corresponding to
$\Symm^5 (f)$, i.e., such that the system of Galois representations attached to
$\pi$ agrees with the 5-th symmetric power of the one attached to f. We also
improve the base change result that we obtained in a previous work: for any
newform f, and any totally real number field F (no extra assumptions on f or
F), we prove the existence of base change relative to the extension $F/\Q$.
Finally, we combine the previous results to deduce that base change also holds
for $\Symm^5(f)$: for any Hecke eigenform f of level 1 and any totally real
number field F, the automorphic form corresponding to $\Symm^5 (f)$ can be base
changed to F.
