Let $ G $ be a perfect group. We often consider extensions

$$

1 \to A \to E \to G \to 1,

$$

where $ A $ is abelian, $ G $ is perfect and the image of $ A $ is central. Is it possible for such an extension not be central? In other words if we have

$$

1 \to A \to E \to G \to 1,

$$

where $ A $ is finite abelian, $ G,E $ finite perfect then must the image of $ A $ be central?

## Best Answer

Let $W := \mathbb{Z}_2 \wr A_5$ and let $G := \operatorname{Inn}(W)$. This is a perfect group of order $960$, has trivial centre, and is of the form $(\mathbb{Z}_2)^4 \rtimes A_5$.

The following GAP code can identiy more examples: