= Solution
Write
$$
\Lambda_4(f_1,f_2,f_3,f_4)
=\mathbb E_{x,d}f_1(x)f_2(x+d)f_3(x+2d)f_4(x+3d).
$$
Apply the <Cauchy-Schwarz inequality> first in the variable carrying $f_1$ and then in the variable carrying $f_2$. After the invertible linear changes of variables permitted by $2,3\nmid|G|$, apply the assumed three-function $U^2$ estimate to the resulting multiplicative derivatives. The standard calculation gives
$$
|\Lambda_4|^8
\leq\|f_1\|_2^8\|f_2\|_2^8
\left(\mathbb E_h\|\partial_hf_3\|_{U^2}^4\right)
\left(\mathbb E_h\|\partial_hf_4\|_{U^2}^4\right).
$$
By the <Derivative identity for the Gowers U3 norm>, the last two factors are $\|f_3\|_{U^3}^8$ and $\|f_4\|_{U^3}^8$. Taking eighth roots proves
$$
|\Lambda_4(f_1,f_2,f_3,f_4)|
\leq\|f_1\|_2\|f_2\|_2\|f_3\|_{U^3}\|f_4\|_{U^3}.
$$
If $A\subseteq G$ has density $\alpha$, write $1_A=\alpha+f$. Expanding $\Lambda_4(1_A,1_A,1_A,1_A)$, the constant term is $\alpha^4$, and the displayed inequality bounds every nonconstant term after translation of one factor by a $U^3$ norm of the balanced function $f$. Thus sufficiently small $\|1_A-\alpha\|_{U^3}$ makes the normalized number of four-term <arithmetic progression>[arithmetic progressions] close to $\alpha^4$.
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