TITLE: Cohomology operations on unoriented cobordism
QUESTION [9 upvotes]: In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law defined by tensor product of bundles). 
Since their properties are so similar to the properties of Steenrod squares, my questions are: do they become normally Steenrod squares in $N^*(X) \otimes_{N*} \mathbb{Z}_2 $? Do they correspond to $Sq^i \otimes id_{N^*} $ under the isomorphism $N^*(X) \simeq H^*(X) \otimes_{\mathbb{Z}_2} N^*$?

REPLY [5 votes]: There is a natural transformation $N^*(X)\to H^*(X;\mathbb{Z}/2)$ given by the universal Thom class, and this maps the Steenrod-tom Dieck operations in cobordism to the Steenrod squares in cohomology. A reference is Section 15 of
Tammo tom Dieck, MR 244989 Steenrod-Operationen in Kobordismen-Theorien, Math. Z. 107 (1968), 380--401.
This answers your first question, I think. As to your second question, I don't know the answer but I suspect the Steenrod-tom Dieck operations are non-trivial on $N^*(pt)$.