2 fixed the error pointed out in the comments

As Willie already said, you should look at the action of your group on the the extreme rays of C. Look at the kernel of this action, K. The group GL(C)/K is then finite (and bounds on its order can be derived from the fact that it will be a permutation action, that is realised in a subspace of certain dimension...)

Regarding K, $K$, there will be two cases:

1) the number a partition of rays equals the dimension d set of the space spanned by C, and then K could scale each extreme ray independently

2) this number is bigger than d; then K can only scale all rays into components $I_1,\dots,I_k$, so that each basis of them simultaneously, i.e. K acts on the linear span of C by (positive) scalar multiplications.

The case 1) is kind $C$ consisting of trivial extreme rays will have $\dim (V_{I_j})$ elements from $V_{I_j}$, the linear span of $I_j$. Then $K$ will induce the multiplication by positive scalars action on each $V_{I_j}$, and GL(C)/K is quite obvious)will be the direct product of these actions.

(Thanks to David Speyer for pointing the error in the original description of $K$).

1

As Willie already said, you should look at the action of your group on the the extreme rays of C. Look at the kernel of this action, K. The group GL(C)/K is then finite (and bounds on its order can be derived from the fact that it will be a permutation action, that is realised in a subspace of certain dimension...)

Regarding K, there will be two cases:

1) the number of rays equals the dimension d of the space spanned by C, and then K could scale each extreme ray independently

2) this number is bigger than d; then K can only scale all of them simultaneously, i.e. K acts on the span of C by (positive) scalar multiplications.

The case 1) is kind of trivial (and GL(C)/K is quite obvious).