Suppose $B$ is an abelian group such that for every integer $n\ge 1$, the $n$-torsion subgroup $B[n]$ is finite.

Let $B_{\rm tor} = \varinjlim_{n\ge 1} B[n]$ be the torsion subgroup of $B$.

Is it true that, necessarily, there exists an integer $d\ge 0$ such that

$$B_{\rm tor} \simeq (\mathbf{Q}/\mathbf{Z})^d\oplus F,$$ for $F$ a finite group?

What if we replace $B_{\rm tor}$ by $B[\ell^{\infty}] = \varinjlim_{n\ge 1} B[\ell^n]$ for a single prime $\ell$, and $(\mathbf{Q}/\mathbf{Z})^d\oplus F$ by $(\mathbf{Q}_{\ell}/\mathbf{Z}_{\ell})^d\oplus F_{\ell}$ for $F_{\ell}$ a finite $\ell$-group?