I don't have a copy of your text but the Bethe approximation considers a cluster of a central spin ##\sigma_0## in an external field h, and two surrounding spins (in the 1-s Ising model), in an effective field h', which mimics the remaining crystalline lattice. The spin Hamiltonian of this cluster is:$$\mathcal H_c=\mathcal J\sigma_0 \left (\sigma_1 + \sigma_2 \right ) - h\sigma_0 -h'\left (\sigma_1 + \sigma_2 \right )$$
and the partition function of this cluster is:$$\mathcal Z_c=\sum_{ \sigma_i } exp\left (-\beta \mathcal H_c \right )$$
The spins in the sum can take the values +1 or -1. Using the notation from your book construct the following table:
##\sigma_0## ##\sigma_1## ##\sigma_2## ##\mathcal Z_c##
## 1## ## 1## ## 1## ## xyz##
## 1## ## -1## ## 1## ## x##
## 1## ## 1## ## -1## ## x##
## 1## ## -1## ## -1## ## xy^{-1}z^{-1}##
## -1## ## 1## ## 1## ## x^{-1}y^{-1}z##
## -1## ## -1## ## 1## ## x^{-1}##
## -1## ## 1## ## -1## ## x^{-1}##
## -1## ## -1## ## -1## ## x^{-1}yz^{-1}##
Having found ##\mathcal Z_c## I assume you can calculate:
$$ < \sigma_0 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h}$$
and
$$ < \sigma_1 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h'}$$