$\Out(A_4)$ is nontrivial

Because \(A_4\trianglelefteq S_4\), conjugation by the odd permutation \((12)\) restricts to an automorphism \[ \varphi:A_4\to A_4. \] We show that \(\varphi\) is not inner.

Suppose there were \(g\in A_4\) such that \[ (12)x(12)^{-1}=gxg^{-1} \qquad\text{for every }x\in A_4. \] Then \(g^{-1}(12)\) would centralize \(A_4\). But \[ C_{S_4}(A_4)\subseteq C_{S_4}((123))\cap C_{S_4}((124)). \] The two centralizers on the right are respectively \[ \langle(123)\rangle \quad\text{and}\quad \langle(124)\rangle, \] whose intersection is trivial. Hence \[ C_{S_4}(A_4)=1. \] Thus \(g^{-1}(12)=1\), forcing \(g=(12)\), impossible because \(g\in A_4\).

Therefore \(\varphi\) is an outer automorphism, so \[ \boxed{\operatorname{Out}(A_4)\ne1}. \]