  Considered are Wiener--Hopf plus Hankel operators
$W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions
$a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost
periodic functions and Fourier images of $L^1(\mathbb{R})$-functions. If the
generating functions $a$ and $b$ satisfy the matching condition
  \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R},
  \end{equation*} an explicit description for the kernels and cokernels of the
operators mentioned is given.
