Let $A$ be a commutative ring and let $M$ be an $A$-module. Let $I$ be an ideal contained in the annihilator of $M$. Then, we know that $M$ is a left $A/I$-module and that $M$ is in fact a $(A, A/I)$-multimodule in the sense of Algebra $I$ by Bourbaki.
Does the $A$-module structure on $M$ coincide with the $A$-module structure on $\pi_*(M)$ corresponding to the canonical ring projection $\pi: A\to A/I$. I recall that $\pi_*(M)$ is the underlying additive group $M$ equipped with the scalar multiplication $a\cdot m:=\pi(a)\cdot m$.