= Solution
{c}
Write $\kappa=R_d/c_p$, where $R_d$ is the <specific gas constant> and $c_p$ the <specific heat capacity at constant pressure>. Conservation of <potential temperature> on a dry <adiabatic process> gives
$$
T(p)=T_g\left(\frac p{p_s}\right)^\kappa,\qquad T_c=qT_g,\qquad q=\left(\frac{p_c}{p_s}\right)^\kappa.
$$
Let $Q=(1-\alpha_c)S$ be the absorbed stellar <irradiance>, and let $T_0=(Q/\sigma)^{1/4}$, with $\sigma$ the <Stefan-Boltzmann constant>. The transparent gas contributes no <thermal radiation>. For a <cloud> with <emissivity> $e$, <Kirchhoff's law of thermal radiation> gives thermal absorptivity $e$; assume thermal transmissivity $1-e$ and negligible thermal reflection, as in the usual one-layer <grey body> model. The outgoing <radiant flux> is the transmitted surface emission plus upward <cloud> emission:
$$
Q=(1-e)\sigma T_g^4+e\sigma T_c^4.
$$
Thus
$$
\boxed{T_g=T_0\left[1-e+eq^4\right]^{-1/4},\qquad T_c=qT_g.}
$$
No factor of four is needed: $S$ is the incident <irradiance> of the specified column, rather than a stellar-beam <irradiance> that still needs a spherical average.
For fixed $\alpha_c,p_c,p_s$, increasing $e$ monotonically warms the ground when $0<q<1$. At $e=0$, $T_g=T_0$; at $e=1$, $T_c=T_0$ and $T_g=T_0/q$. A <cloud> at the surface, $q=1$, produces no such warming. Comparing with a genuinely cloud-free column also requires including its different <albedo>: relative to $(S/\sigma)^{1/4}$, the ratio is $[(1-\alpha_c)/(1-e+eq^4)]^{1/4}$. Reflection can outweigh the <greenhouse effect>.
The governing balance here is the whole-column <radiation budget>. Imposing an additional purely radiative surface balance would incorrectly remove the <convective heat transfer> that the prescribed dry <adiabatic process> may require.
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