Variant of assignment problem
You want to assign $n$ persons to $m$ groups, so that

Each person is assigned one group.
Each group is assigned at most two people.
For each possible person $i$ and group $j$ there is a score $C_{ij}$, and you want to maximize the sum of scores of all assignments.

If such an assignment is at all possible, then $n \leq 2m$.
Create an instance of the usual assignment problem (in which we are looking for a one-to-one correspondence between persons and groups) as follows:

There are $n$ real persons $1,\ldots,n$ and $2m-n$ dummy persons $n+1,\ldots,2m$.
There are $2m$ groups, composed of two copies of the original groups: $1,\ldots,m$ and $m+1,\ldots,2m$.
The costs $C'_{ij}$ of the instance are: for $i \in \{1,\ldots,n\}$ and $j \in \{1,\ldots,m\}$, define $C'_{ij} = C'_{i(j+m)} = C_{ij}$, and set all other costs to zero.

Every feasible solution of the original problem translates to a feasible solution of the new problem with identical cost, and vice versa.