The above image is from the book of Dummit and Foote, Edition 3. In the fourth paragraph, the authors claim that "equivalences involving the same extension module $ B $ are automorphisms of $ B $ that restrict to identity map on both $ \psi (A) $ and $ B / \psi(A) $".
My question is what claim the authors are trying to make? If the claim is that if we have two short exact sequences,
$$ 0 \longrightarrow A \stackrel{\psi}{ \longrightarrow} B \stackrel{ \phi_{1} } { \longrightarrow } C \longrightarrow 0 $$
$$ 0 \longrightarrow A \stackrel{\psi}{ \longrightarrow} B \stackrel{ \phi_{2} } { \longrightarrow } C \longrightarrow 0 $$
which are equivalent to each other, then the automorphism $ \beta : B \longmapsto B $ must be such that $ \beta( \psi(a)) = \psi(a) $ and $ \beta( b ) + \psi(A) = b + \psi(A) $ for all $ b \in B $, then, this wrong unless $ \phi_{1} = \phi_{2} $, since, we have
$$ \beta(b) + \psi(A) = b + \psi(A) \implies \beta(b) - b \in \psi(A) = \ker \phi_{1} = \ker \phi_{2} \implies \phi_{2} ( \beta(b) - b) = 0 \implies \phi_{1}(b) = \phi_{2}(b) . $$
Also, the next sentence doesn't seem to be smoothly connected to the first one. In my opinion, if you delete the first sentence, the text makes perfect sense. So, what did the authors had in mind when they made that claim?