Let $K$ be a local field, complete with respect to a discrete valuation $v$ and let $E/K$ be an elliptic curve. We also let $E_0(K)$ is the set of points with nonsingular reduction.

It is known that $E(K)/E_0(K)$ is of order at most 4 if $E$ does not have split multiplicative reduction over $K$ (see Theorem VII.6.4 of Silverman's "The Arithmetic of Elliptic Curves").

My question concerns the case when the order of this group is exactly 4. How can we tell if this group is ${\mathbb Z} / {2\mathbb Z} \times {\mathbb Z} / {2\mathbb Z}$ or ${\mathbb Z} / {4\mathbb Z}$?

I looked at Tate's Algorithm, as described in Section IV.9 of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", and although that tells us how to tell when the order is 4, and Table 4.1 states what the group is when the residue field, $k$, is algebraically closed, it does not tell us how to determine the actual group itself.

Any references, ideas, etc would be very welcome.