$\newcommand\Ext{\mathrm{Ext}} \newcommand\Z{\mathbb{Z}} \newcommand\G{\mathbb{G}}$ The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-local Moore spectrum at the prime 3 revisted" (so everything is at $n=2,p=3$).

Let $\G_2$ be the extended Morava stabiliser group, and let $\G_2^1$ be the kernel of the reduced norm $\G_2 \to \Z_3$. Then Henn,Karamanov and Mahowald use a permutation resolution of $\mathbb{G}_2^1$ to construct a spectral sequence (the algebraic spectral sequence)

$$ E_1^{p,q,t}=\Ext_{\mathbb Z_3[[\G_2^1]]}^q(C_p,M_t)\Longrightarrow H^{p+q}(\G_2^1,M_t) $$ for $M=(E_2)_*(V(0)) = (E_2)_*/(3)$ and then use this to calculate $H^*(\G_2,(E_2)_*/(3)) $, which in turn is used to calculate $\pi_*(L_{K(2)}V(0))$ (via the Adams-Novikov spectral sequence)

It's a very technical paper, and suffice to say I can't define all the terms I'm going to use in an introduction. But I am specifically interested in some of the differentials in the Adams-Novikov spectral sequence. In particular, let's just start with the first differential $d_5(\Delta_k \tilde{\alpha} \beta) = \pm \Delta_{k-1}\beta^4 v_1$. (The calculation is on pp. 32-33)

Consider the short exact sequence

$$ 0 \to \Sigma^4 (E_2)_*/(3) \stackrel{v_1}{\to} (E_2)_*/(3) \to (E_2)_*(3,u_1) \to 0$$

It turns out that $\Delta_k \tilde{\alpha} \beta$ is in the kernel of multiplication by $v_1$ and so must be in the image of the Bockstein $\delta_{\G_2}^1$ in $H^*(\G_2,-)$ and $\delta_{\G_2^1}^1$ in $H^*(\G_2^1,-)$ associated to the above short exact sequence (after appropriate suspension). So I go and look up the calculation of $H^*(\G_2^1,(E_2)_*/(3,u_1))$ and work out (by degree) what possible classes it can be to get that $$\delta_{\G_2^1}^1((\omega^2 u^{-4})^{3k+2}\beta) = \pm \Sigma^4 \Delta_k \tilde{\alpha}\beta,$$

and in fact the same is true for $\delta^1_{\G_2^1}$. What I am interested in is how the geometric boundary theorem is then used to get the result. From Ravenel (2.3.4) this applies to a cofiber sequence $$W \stackrel{f}{\to} X \stackrel{g}{\to} Y \stackrel{h}{\to} \Sigma W$$ with $E_*(h)=0$. If such a situation applies then we have maps $\delta_r$ for $2 \le r \le \infty$ such that $\delta_2 = \delta^1$ in our language (I think), but more importantly $\delta_r d_r = d_r \delta_r$.

Thus I think the calculation of $d_5(\Delta_k \tilde{\alpha} \beta)$ should basically be something akin to $d_5(\Delta_k \tilde{\alpha} \beta) = d_5\delta(-) = \delta d_5(-)$, where $d_5(-)$ is something we can calculate via knowledge of differentials in $H^*(\G_2,(E_*)/(3,u_1)).$

Is this the right idea? I am slightly confused by the $\Sigma^4$ floating around and more confused by the fact that HKM seem to be doing some additional calculations (e.g. why do I care about $\delta^1((\omega^2 u^{-4})^{3k+2} \beta)$?)