Commutativity among the elements in the quotient group $G$ and the ful group $H$ under homomorphism $H \overset{r}{\to} G$ Say the full group is $H$, and we pick up a normal subgroup $N$, and we define the quotient group $$G=H/N.$$
There is a group homomorphism $r$ from $H$ to $G$
$$
H \overset{r}{\to} G.
$$
My question concerns the following relations are true in general or whether there are counter examples?


*

*$ g \cdot r(h) \cdot g^{-1}=r(h)$,  for $\forall h \in H$, $\forall g \in G$.

*$ r(h) \cdot g \cdot r(h)^{-1}=g$, for $\forall h \in H$, $\forall g \in G$.
If some of them is not true, please give the simplest (counter) example (especially the finite group).

They do not hold in general.
Consider $H= D_8 = <\sigma, \tau | \sigma^8 = 1, \tau^2 = 1, \tau \sigma^i \tau = \sigma^{8-i}>$ and consider its normal subgroup $N=<1, \sigma^4>$.
Now $G=H/N$ has order $8$ and has elements $1, \tau, \sigma, \sigma^2, \sigma^3, \sigma\tau, \sigma^2 \tau,\sigma^3 \tau$. 
Consider $r:H \to G$ the usual projection. Now $g=\tau, h=\sigma^5\tau$ gives you
$$\tau r(\sigma^5\tau) \tau = r(\tau\sigma^5) = \tau\sigma$$
while $r(\sigma^5 \tau) = \sigma\tau$. You can do the same for the other relation.
