Classification of linear mappings between indefinite inner product spaces

Let $\mathcal A:U\to V$ be a linear mapping between vector spaces $U$ and $V$ over a field or skew field $\mathbb F$ with symmetric, or skew-symmetric, or Hermitian forms $\mathcal B:U\times U\to\mathbb F$ and $\mathcal C:V\times V\to\mathbb F.$ We classify the triples $(\mathcal A,\mathcal B,\mathcal C)$ if $\mathbb F$ is $\mathbb R$, or $\mathbb C$, or the skew field of quaternions $\mathbb H$. We also classify the triples $(\mathcal A,\mathcal B,\mathcal C)$ up to classification of symmetric forms and Hermitian forms if the characteristic of $\mathbb F$ is not 2.