What is a Mackey Obstruction?
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I think, it is the obstruction of extending a projective representation of a group to a linear one considered as an element of some H^2 cohomology group. This is quite standard and classical. You may find the definition in e.g. :
Curtis, Charles W.; Reiner, Irving Representation theory of finite groups and associative algebras. Reprint of the 1962 original. AMS Chelsea Publishing, Providence, RI, 2006.
Curtis, Charles W.; Reiner, Irving Methods of representation theory. Vol. I. With applications to finite groups and orders. Reprint of the 1981 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1990. xxiv+819 pp. ISBN: 0-471-52367-4, 20-02
For a succinct answer, read a bit of Section 2 of this paper.
Also, there is a survey (the article by Judith Packer) of projective representations and the Mackey obstruction. "Projective representations and the Mackey obstruction - a survey", Contemporary Mathematics, v. 449 (2008), pp. 345-378.
(I posted this same reference as an answer to another question this morning, and am surprised that both of these questions weren't asked by the same person. You two should get together and talk...)