I'm having trouble with the following simple exercise from p.9 of Brinon-Conrad's p-adic Hodge theory notes. The question, if I've identified $\Lambda/p^n\Lambda$ correctly, is to show $$Aut_\mathbb{Z}(\mathbb{Z}/p^n\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}) \to Aut_\mathbb{Z}(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z})$$ is not surjective (I guess for $n>1$).
Can anyone give hints? The maps seem surjective to me, because we can easily lift a matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ to one in $\mathbb{Z}/p^n\mathbb{Z}$, and the invertibility requirement in $\mathbb{Z}/p^n\mathbb{Z}$ is just that determinant is nonzero in $\mathbb{Z}/p\mathbb{Z}$.
Here's my writeup, along with the problem statement. $\mathbb{Z}_p$ denotes the $p$-adic integers.
What about the automorphism $\alpha$ of $\newcommand{\Z}{\Bbb Z}\Z/p\Z\oplus \Z/p\Z$ interchanging the two factors? If $\pi:\Z/p^2\Z\oplus\Z/p\Z \to \Z/p\Z\oplus\Z/p\Z$ is the projection, then can there be an automorphism $\beta$ of $\Z/p^2\Z\oplus\Z/p\Z$ reducing to $\alpha$. What would $\beta(0,1)$ be? It must be $(c,0)$ where $p\nmid c$ and $c\in \Z/p^2\Z$, but that has order $p^2$.