This pertains to BBFSK. In the current context, a complex of a group is simply a subset of the group's set. The group under discussion is $\mathfrak{G}.$
In 1B2 $\S$ 3.1 (page 183) we are told:
For a given complex $\mathfrak{K},$ the complexes of the form $G^{-1}\mathfrak{K}G,G\in\mathfrak{G}$ are called the conjugates or transforms of $\mathfrak{K}$ (under $\mathfrak{G}$). If $\mathfrak{K}$ is conjugate only to itself, then $\mathfrak{K}$ is said to be normal or invariant in $\mathfrak{G}.$
First of all, they don't explain what it means for one complex to be conjugate to another, or to itself. I assume this means, for example, $G^{-1}\mathfrak{K}G=\mathfrak{K};$ but I'm not certain if it means this holds for all $G\in\mathfrak{G},$ or for some, possibly proper subset of $\mathfrak{G}$.
So my first question is, what does it mean to say two complexes are conjugate?
Then in $\S$ 3.4 (page 185) we find:
If $\mathfrak{U}$ is a subgroup of $\mathfrak{G}$, the conjugate complexes $\mathfrak{G}^{-1}G,G\in\mathfrak{G}$ are also subgroups of $\mathfrak{G}.$
Is this a typographical error, and perhaps it should be $G^{-1}\mathfrak{K}G,G\in\mathfrak{G}$ rather than $\mathfrak{G}^{-1}G,G\in\mathfrak{G}?$ If not then what does the statement mean?
The discussion almost certainly relates to what are sometimes called similarity transformations, in the context of matrices, eigenvectors and eigenvalues. Since I am somewhat familiar with such applications, an example along those lines might be helpful.
The first equation in $\S$IB2.6.1 (Page 194) apparently clarifies this. They do intend that $G^{-1}\mathfrak{K}G=\mathfrak{K}$ holds for all $G\in\mathfrak{G}.$
I also believe $\mathfrak{G}^{-1}G,G\in\mathfrak{G}$ is a typo. Though it may be some cryptic sideways way of saying what they claim it means.
I have not given the intervening discussion rigorous consideration, so I am going to leave this open in case someone wishes to challenge my conclusion.