Induced Action on dual space: confusing in definition Notation: maps will be written on left of argument. [like $f(x)$, so $fg(x)=f(g(x))$].
Let $G$ be a group acting on $V$ (from left) so that we have $\rho:G\rightarrow GL(V)\cong GL_n(F)$.
I wanted to define induced action on $V^*$. 
Following wiki we have : for $\phi\in V^*$ and $v\in V$ written as column vectors, operation of $V^*$on $V$ is 
$$\langle \phi,v\rangle=\phi^Tv \mbox{ (product of row vector by column vector)}.$$
Wiki (or Fulton-Harris) say that the induced representation $\rho^*$ on dual space $V^*$ should be defined in such a way that we should have consistency:
$$\langle \rho^*(\phi), \rho(v) \rangle=\langle \phi,v\rangle.$$
Question 1. I don't understand what is the need of consistency (last equation).
Question 2. Given $\rho:G\rightarrow GL(V)$ a representation of $G$, if $\rho^*:G\rightarrow GL(V^*)$ satisfies above consistency, then I want to deduce that 
$$\rho^*(g)=\rho(g^{-1})^*.$$
How to do this?
Here $\rho(G^{-1})^*$ is understood as follows: if $T:V\rightarrow W$ is a linear transformation then it induces a linear transformation on duals $T^*:W^*\rightarrow V^*$, $T^*(f)=f\circ T$. So for linear transformation $\rho(g^{-1})$ its dual transformation is $\rho(g^{-1})^*$. I know that $(T\circ S)^*=S^*\circ T^*$.

(I was trying to do exercise (2) with help of some commutative diagrams, but I confused too in above complexity; can one make it clear and elegant; one may put hints or small-small exercises instead of answers)
 A: Answer 1: Usually, there can be many actions of $G$ on the dual space $V^*$. It is a good starting point if you require your action on the dual space to have some connection to the original action, and in many cases this type of definition will help you translate problems from the space $V$ to its dual.
This is a very general phenomenon in math. Think for example on the definition of a vector space - it has the structure of an additive group, and also there is a field that acts on it. The structure of the vector space starts to make sense only when you assume that these two structures interrelate using the distributive axiom. Other such examples are rings with the distributive axiom which connects the multiplicative and additive operations, the real numbers with their algebraic operations of addition and multiplication being continuous which is a topological property and you can find many more examples.
Answer 2: both $\rho^*(g)$ and $\rho(g^{-1})^*$ are linear transformation on the dual space $V^*$, so the equality between them is equivalent to $\rho^*(g)(\phi)=\rho(g^{-1})^*(\phi)$ for any functional $\phi$ which is equivalent (by the definition of the dual space) to
$$\langle\rho^*(g)(\phi),v\rangle=\langle\rho(g^{-1})^*(\phi),v\rangle.$$
Try to show that this is true.
