Cardinal nonabsoluteness in a small transitive model

Let $M$ be a transitive set of cardinality $\kappa$ that models enough set theory and contains $\kappa$. The internal successor cardinal $\alpha =({\kappa }^{+}{)}^M$ is an ordinal in $M$, so transitivity gives $\alpha \subseteq M$ and hence $|\alpha |\le \kappa$ externally. Although $M$ regards $\alpha$ as a cardinal number, an ambient rank containing a bijection between $\kappa$ and $\alpha$ does not. Cardinalhood can therefore fail to be absolute even between transitive models.