  In this article we give self-contained existence proofs for the sporadic
simple groups Co_2 and Fi_{22} using the second author's algorithm [10]
constructing finite simple groups from irreducible subgroups of GL_n(2). These
two sporadic groups were originally discovered by J. Conway [4] and B. Fischer
[7], respectively, by means of completely different and unrelated methods. In
this article n=10 and the irreducible subgroups are the Mathieu group M_{22}
and its automorphism group Aut(M_{22}). We construct their five non-isomorphic
extensions E_i by the two 10-dimensional non-isomorphic simple modules of
M_{22} and by the two 10-dimensional simple modules of A_{22} = Aut(M_{22})
over F=GF(2). In two cases we construct the centralizer H_i = C_{G_i}(z_i) of a
2-central involution z_i of E_i in any target simple group G_i. Then we prove
that all the conditions of Algorithm 7.4.8 of [11] are satisfied.
  This allows us to construct G_3 $\cong$ Co_2 inside GL_{23}(13) and G_2
$\cong$ Fi_{22} inside GL_{78}(13). We also calculate their character tables
and presentations.
