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Theorem 2. Let $G$ be a non-trivial group satisfying both chain conditions on normal subgroups. Write $G= (A_1\times\cdots\times A_k)\times (G_1\times \cdots \times G_n)$, and $A=A_1\times\cdots \times A_k$, where the $A_i$ are indecomposable abelian groups and the $G_i$ are indecomposable non-abelian groups. If $G$ is a $\mathcal{D}$-group then the following three conditions hold:

 

(1) The Sylow subgroups of $A$ are either cyclic or elementary abelian. Equivalently, for all $i\neq j$, any non-trivial element of $\text{Hom}(A_i,A_j)$ is an injection.

 

(2) The Krull-Schmidt decomposition of $G_1\times\cdots\times G_n$ is unique, up to the order of the factors. Equivalently, for all $i\neq j$, $\text{Hom}(G_i,Z(G_j))$ is trivial.

 

(3) For all $i,j$, any non-trivial element of $\text{Hom}(A_i,Z(G_j))$ is an injection. Given the previous two conditions, this is equivalent to saying that if the Sylow $p$-subgroup of $A$ is not elementary abelian, then the Sylow $p$-subgroup of $Z(G_j)$ is trivial for all $j$.

Theorem 2. Let $G$ be a non-trivial group satisfying both chain conditions on normal subgroups. Write $G= (A_1\times\cdots\times A_k)\times (G_1\times \cdots \times G_n)$, and $A=A_1\times\cdots \times A_k$, where the $A_i$ are indecomposable abelian groups and the $G_i$ are indecomposable non-abelian groups. If $G$ is a $\mathcal{D}$-group then the following three conditions hold:

(1) The Sylow subgroups of $A$ are either cyclic or elementary abelian. Equivalently, for all $i\neq j$, any non-trivial element of $\text{Hom}(A_i,A_j)$ is an injection.

(2) The Krull-Schmidt decomposition of $G_1\times\cdots\times G_n$ is unique, up to the order of the factors. Equivalently, for all $i\neq j$, $\text{Hom}(G_i,Z(G_j))$ is trivial.

(3) For all $i,j$, any non-trivial element of $\text{Hom}(A_i,Z(G_j))$ is an injection. Given the previous two conditions, this is equivalent to saying that if the Sylow $p$-subgroup of $A$ is not elementary abelian, then the Sylow $p$-subgroup of $Z(G_j)$ is trivial for all $j$.

(edit: Added Theorem 2 below, which gives half the case with abelian direct factors, and classifies the finite abelian $\mathcal{D}$-groups)

Theorem Let $G$ be a non-trivial group satisfying both chain conditions on normal subgroups, and with no non-trivial abelian direct factors. Then $G$ is a $\mathcal{D}$-group if and only if $G$ admits a unique Krull-Schmidt decomposition, up to the order of the factors.

The reverse direction did not use the assumption of no abelian direct factors, and effectively covers all examples in the OP and Keith's answer, and I'm fairly sure we can readily handle their presence in the forward direction. You can rule out a lot of cases by observing when we can exhibit a non-trivial $z$ with $\ker(z)\neq 1$. Also note how the exact idea for the forward direction also yields the OP's example of $\mathbb{Z}_4\times \mathbb{Z}_2$ by using the non-trivial $z\in\text{Hom}(\mathbb{Z}_4,\mathbb{Z}_2)$. But I've not yet worked out the details on this case, otherwise.

Theorem 1. Let $G$ be a non-trivial group satisfying both chain conditions on normal subgroups, and with no non-trivial abelian direct factors. Then $G$ is a $\mathcal{D}$-group if and only if $G$ admits a unique Krull-Schmidt decomposition, up to the order of the factors.

The reverse direction did not use the assumption of no abelian direct factors.

Theorem 2. Let $G$ be a non-trivial group satisfying both chain conditions on normal subgroups. Write $G= (A_1\times\cdots\times A_k)\times (G_1\times \cdots \times G_n)$, and $A=A_1\times\cdots \times A_k$, where the $A_i$ are indecomposable abelian groups and the $G_i$ are indecomposable non-abelian groups. If $G$ is a $\mathcal{D}$-group then the following three conditions hold:

(1) The Sylow subgroups of $A$ are either cyclic or elementary abelian. Equivalently, for all $i\neq j$, any non-trivial element of $\text{Hom}(A_i,A_j)$ is an injection.

(2) The Krull-Schmidt decomposition of $G_1\times\cdots\times G_n$ is unique, up to the order of the factors. Equivalently, for all $i\neq j$, $\text{Hom}(G_i,Z(G_j))$ is trivial.

(3) For all $i,j$, any non-trivial element of $\text{Hom}(A_i,Z(G_j))$ is an injection. Given the previous two conditions, this is equivalent to saying that if the Sylow $p$-subgroup of $A$ is not elementary abelian, then the Sylow $p$-subgroup of $Z(G_j)$ is trivial for all $j$.

proof: The proof proceeds essentially as in proof of the forward direction for the previous proof.

We write $G=H_1\times\cdots \times H_m$, where we permit one or more factors to be abelian. For any $i\neq j$ we consider $z\in\text{Hom}(H_i,Z(H_j))$, and define $\phi\in\text{Aut}_c(G)$ by $\phi(g)=g$ for all $g\in H_k\neq H_i$ and $\phi(g)=g z(g)$ for all $g\in H_i$. We consider $H_i\cap \phi(H_i)=\ker(z)$, which is the intersection of two direct factors of $G$.

If $G$ is a $\mathcal{D}$-group, then $\ker(z)$ is a normal subgroup of the indecomposable group $H_i$ which is also a direct factor of $G$. Therefore either $\ker(z)=H_i$ or $\ker(z)=1$. In particular, for all $i\neq j$ any non-trivial element of $\text{Hom}(H_i,Z(H_j))$ must be an injection. If such an injection exists, then $H_i$ must be abelian. As every quotient of a finite abelian group $X$ is isomorphic to a subgroup of $X$, and vice versa, the three conditions then follow. $\square$.

I'm fairly confident the converse also holds, thereby giving the full classification of $\mathcal{D}$-groups with both chain conditions, and so in particular all finite $\mathcal{D}$-groups. But I'm still working on that.