  We classify possible finite groups of symplectic automorphisms of K3 surfaces
of order divisible by 11. The characteristic of the ground field must be equal
to 11. The complete list of such groups consists of five groups: the cyclic
group of order 11, $11\rtimes 5$, $L_2(11)$ and the Mathieu groups $M_{11}$,
$M_{22}$. We also show that a surface $X$ admitting an automorphism $g$ of
order 11 admits a $g$-invariant elliptic fibration with the Jacobian fibration
isomorphic to one of explicitly given elliptic K3 surfaces.
