Let $K$ be a finite extension of $\mathbb{Q}$. Let $T$ be a finite dimensional $G_{K}$ module over $\mathbb{Z}_p$. Does the Bloch-Kato Selmer group $H^{1}_{f}$ satisfy $H^1_f(\mathbb{Q},Ind^{G_{\mathbb{Q}}}_{G_K}T) = H^{1}_{f}(K,T) $?
If not, then can you tell me what kind of condition on $K/\mathbb{Q}$ as well as the condition at the prime $p$ do I need for that equality to hold? Thank you. (I was trying to use $Ind(Res V\otimes W)\simeq V\otimes Ind W$ for each prime but I got stuck.)
This is a nice exercise. You start by making some reduction steps. Firstly, the BK Selmer group for $T$ is by definition saturated (i.e. it is the preimage in $H^1(K, T)$ of a $\mathbf{Q}_p$-subspace of $H^1_{\mathrm{f}}(K, T \otimes \mathbf{Q}_p)$) so you can invert $p$ everywhere and ask the question for finite-dimensional $\mathbf{Q}_p$-linear representations $V$
Secondly, since you know Shapiro's lemma for the global cohomology, it is sufficient to check that the local conditions match, i.e. that for each prime we have $$H^1_{\mathrm{f}}(\mathbf{Q}_\ell, \operatorname{Ind}_K^{\mathbf{Q}} V) = \bigoplus_{v \mid \ell} H^1_{\mathrm{f}}(K_v, V).$$
The next reduction step is to use Mackey's theorem to write $(\operatorname{Ind}_K^{\mathbf{Q}} V) |_{\mathbf{G}_{\mathbf{Q}_\ell}}$ as a direct sum of pieces indexed by the quotient $G_{\mathbf{Q}_{\ell}} \backslash G_\mathbf{Q} / G_K$. This double quotient turns out to naturally biject with the set of primes of $K$ above $\ell$, and the summand corresponding to the prime $v$ is $\operatorname{Ind}_{G_{K_v}}^{G_{\mathbf{Q}_\ell}}\left( V_{G_{K_v}}\right)$. So now we are down to a purely local question: if $M/L$ is a finite extension of $\ell$-adic fields for some $\ell$, we need to check that $$ H^1_{\mathrm{f}}(L, Ind_{G_{M}}^{G_{L}} V) = H^1_{\mathrm{f}}(M, V).$$ This I gather you already know how to do.
