  Using density functional theory we investigate the occurrence of
superconductivity in AC$_6$ with A=Mg,Ca,Sr,Ba. We predict that at zero
pressure, Ba and Sr should be superconducting with critical temperatures
(T$_c$) 0.2 K and 3.0 K, respectively. We study the pressure dependence of
T$_c$ assuming the same symmetry for the crystal structures at zero and finite
pressures. We find that the SrC$_6$ and BaC$_6$ critical temperatures should be
substantially enhanced by pressure. On the contrary, for CaC$_6$ we find that
in the 0 to 5 GPa region, T$_c$ weakly increases with pressure. The increase is
much smaller than what shown in several recent experiments. Thus we suggest
that in CaC$_6$ a continous phase transformation, such as an increase in
staging, occurs at finite pressure. Finally we argue that, although MgC$_6$ is
unstable, the synthesis of intercalated systems of the kind
Mg$_x$Ca$_{1-x}$C$_y$ could lead to higher critical temperatures.
