Fiber bundle and fibration of classifying space Let $BG$ is classifying space of $G$ topological group.
If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the
inclusion map $i:H\rightarrow G$ induces 
\begin{equation*}
G/H\rightarrow BH\rightarrow BG
\end{equation*}
a fiber bundle?
If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the
inclusion map $i:H\rightarrow G$ induces
\begin{equation*}
G/H\rightarrow BH\rightarrow BG
\end{equation*}
a fibration?
If $G$ is any compact group and $N$ is a closed normal subgroup of $G$, then
the quotient map $\pi :G\rightarrow G/N$ induces
\begin{equation*}
BN\rightarrow BG\rightarrow B\left( G/N\right) 
\end{equation*}
a fiber bundle?
If $G$ is any compact group and $H$ is a closed normal subgroup of $G$, then
the quotient map $\pi :G\rightarrow G/N$ induces 
\begin{equation*}
BN\rightarrow BG\rightarrow B\left( G/N\right) 
\end{equation*}
a fibration?

Hint: I assume $H$ is closed here. if $G$ is a Lie group and $H$ a CLOSED subgroup of $G$, the classifying space of $G$ is base space of the universal bundle $p_G:EG\rightarrow BG$. Remark that $EG$ is characterized by the fact that it is contractible. Thus the quotient of $EH$ by $H$ is the classifying space of $H$. This gives you a fibration $G/H\rightarrow EG/H=BH\rightarrow EG/G= (EG/H)/G= BG$. 
You can apply a similar method to the other questions.
