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In the presence of a nonzero B-field, the symmetries of the $\mathrmE_8\times \mathrm E_8$ heterotic string form a 2-group, or a categorifiedgroup, as do the symmetries of the CHL string. We express the bordism groups ofthe corresponding tangential structures as twisted string bordism groups, thencompute them through dimension 11 modulo a few unresolved ambiguities. Then, weuse these bordism groups to study anomalies and defects for these two stringtheories.

Bordism for the 2-group symmetries of the heterotic and CHL strings