adams operations - Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113

Warning: I'll be using the "pre-$lambda$-ring" and "$lambda$-ring" nomenclature, as opposed to the "$lambda$-ring" and "special $lambda$-ring" one (although I just used the latter a few days ago on MO). It's mainly because both sources use it, and I am (by reading them) slowly getting used to it.

Let $G$ be a finite group. The Burnside ring $Bleft(Gright)$ is defined as the Grothendieck ring of the category of finite $G$-sets, with multiplication defined by cartesian product (with diagonal structure, or at least I have difficulties imagining any other $G$-set structure on it; please correct me if I am wrong).

For every $ninmathbb{N}$, we can define a map $sigma^n:Bleft(Gright)to Bleft(Gright)$ as follows: Whenever $U$ is a $G$-set, we let $sigma^n U$ be the set of all multisets of size $n$ consisting of elements from $U$. The $G$-set structure on $sigma^n U$ is what programmers call "map": an element $gin G$ is applied by applying it to each element of the multiset. This way we have defined $sigma^n U$ for every $G$-set $U$; we extend the map $sigma^n$ to all of $Bleft(Gright)$ (including "virtual" $G$-sets) by forcing the rule

$displaystyle sigma^ileft(u+vright)=sum_{k=0}^isigma^kleft(uright)sigma^{i-k}left(vright)$ for all $u,vin Bleft(Gright)$.

Ah, and $sigma^0$ should be identically $1$, and $sigma^1=mathrm{id}$. Anyway, this works, and gives a "pre-$sigma$-ring structure", which is basically the same as a pre-$lambda$-ring structure, with $lambda^i$ denoted by $sigma^i$. Now, we turn this pre-$sigma$-ring into a pre-$lambda$-ring by defining maps $lambda^i:Bleft(Gright)to Bleft(Gright)$ by

$displaystyle sum_{n=0}^{infty}sigma^nleft(uright)T^ncdotsum_{n=0}^{infty}left(-1right)^nlambda^nleft(uright)T^n=1$ in $Bleft(Gright)left[left[Tright]right]$ for every $uin Bleft(Gright)$.

Now, let me quote two sources:

Donald Knutson, $lambda$-Rings and the Representation Theory of the Symmetric Group, 1973, p. 107: "The fact that $Bleft(Gright)$ is a $lambda$-ring and not just a pre-$lambda$-ring - i. e., the truth of all the identities - follows from [...]"

Michiel Hazewinkel, Witt vectors, part 1, 19.46: "It seems clear from [370] that there is no good way to define a $lambda$-ring structure on Burnside rings, see also [158]. There are (at least) two different choices giving pre-$lambda$-rings but neither is guaranteed to yield a $lambda$-ring. Of the two the symmetric power construction seems to work best."
(No, I don't have access to any of these references.)

For a long time I found Knutson's assertion self-evident (even without having read that far in Knutson). Now I tend to believe Hazewinkel's position more, particularly as I am unable to verify one of the relations required for a pre-$lambda$-ring to be a $lambda$-ring:

$lambda^2left(uvright)=left(lambda^1left(uright)right)^2lambda^2left(vright)+left(lambda^1left(vright)right)^2lambda^2left(uright)-2lambda^2left(uright)lambda^2left(vright)$ for $Bleft(Gright)$.

What also bothers me is Knutson's "conjecture" on p. 113, which states that the canonical (Burnside) map $Bleft(Gright)to SCFleft(Gright)$ is a $lambda$-homomorphism, where $SCFleft(Gright)$ denotes the $lambda$-ring of super characters on $G$, with the $lambda$-structure defined via the Adams operations $Psi^nleft(varphileft(Hright)right)=varphileft(H^nright)$ (I think he wanted to say $left(Psi^nleft(varphiright)right)left(Hright)=varphileft(H^nright)$ instead) for every subgroup $H$ of $G$, where $H^n$ means the subgroup of $G$ generated by the $n$-th powers of elements of $H$. This seems wrong to me for $n=2$ and $H=left(mathbb Z / 2mathbb Zright)^2$ already. And if the ring $Bleft(Gright)$ is not a $lambda$-ring, then this conjecture is wrong anyway (since the map $Bleft(Gright)to SCFleft(Gright)$ is injective).

Can anyone clear up this mess? I am really confused...

Thanks a lot.