(Real)linear preservers of multiples of unitaries and matrix pairs with some extremal norm properties

We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs satisfying the extremal norm properties $\mathrm{\parallel }AB\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$, $\mathrm{\parallel }{A}^{\ast }B\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$, or $\mathrm{\parallel }A{B}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$, for the spectral norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$.