**Edit:** According to the essential comment of Alex Degtyarev, we revise the question as follows;

Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $O(m)$, respectively. Is there an oriention preserving automorphism of $O(n+m)$ which restriction to $O(n)\oplus O(m)$ equal to $\alpha \oplus \beta$?

One can repeat the same question for complex version,i.e:$U(n)$.

The reason for consideration of "oriention preserving" is that: It is unlike that the answer would be positive for $\alpha(z)=z$ and $\beta(w)=\bar{w}$, as automorphisms of $U(1)$.

The reason for consideration of $O(n)$, instead of $GL(n,\mathbb{R})$ is that: it is unlike that the answer would be positive for $\alpha(x)=x^{3},\;\beta(x)=x$, as automorphism of $GL(1,\mathbb{R})$.

**Our main motivation:**

We try to define a (possible) equivalent relation on Riemannian vector bundles as follows:

Assume that $E$ and $F$ are two vector bundles over $X$.

**We say that $E$ is equivalent to $F$ if there are cocycles $g_{\alpha\beta}$ and $h_{\alpha\beta}$, respectively for $E$ and $F$ and an (oriention preserving) automorphism $\lambda$ of $O(n)$ such that $g_{\alpha\beta}=\lambda\circ h_{\alpha\beta}$.**

Is this really an equivalent relation?(Transitive property?) Is there a sheaf theoretical language for this question(for this relation)?

Of course, every two ordinary isomorphic bundles, are equivalent in this definition.

If the answer is yes, We wish to define the direct product on this structure (on the space of all equivalent class of Riemannian vector bundles) to obtain a semigroup. So the above question is needed for "well define-ness". Then we consider the Grothendieck group of this semigroup. So we would have a functor $\tilde{\tilde{K}}(X)$.

Now perhaps a natural question is that what would be an appropriate version of Atiyah Janich theorem, here?

As Alex Degtyarev commented on previous version of this post, we ask that how can we define an appropriate higher order of $\tilde{\tilde{K}}$?