I have the following question:

1. Does differential invariants have the same Lie symmetries?

2. I want to know about the relation of differential invariants of an action and their symmetry properties. Does differential invariants of a group action have isomorphic Lie symmetry group?

In [[1]], the authors considered the differential invariants of a fifth-order KdV types of equation (Kawaha KdV equation). The Kawaha KdV equation has the following Lie symmetries:

$$ X_1=\frac{\partial}{\partial x},\qquad X_2=\frac{\partial}{\partial t},\qquad X_3=at\frac{\partial}{\partial x}+\frac{\partial}{\partial u}$$


 The frst-order prolongation of the operator $X$ has the form

$$X^{(3)}=(c_3 at + c_1)\frac{\partial}{\partial x} + c_2 \frac{\partial}{\partial t}+c_3\frac{\partial}{\partial u}-c_3 au_x\frac{\partial}{\partial u_t}-c_3 a u_{xx}\frac{\partial}{\partial u_{xt}}-2c_3 a u_{xt} \frac{\partial}{\partial u_{tt}}-c_3 a u_{xxx}\frac{\partial}{\partial u_{xxt}}-2c_3 a u_{xxt}\frac{\partial}{\partial u_{xtt}}-3c_3 a u_{xtt} \frac{\partial}{\partial u_{ttt}}$$

In the paper a set of generating differential invariants is found. For instance $u_{xxx}$ is a differential invariant of Kawaha KdV. 

Making simple calculations, one obtains

$$X^{(3)}(u_{xxx}) = 0.$$

Therefore $u_{xxx}$ is a differential invariant. However, $u_{xxx}$ has the following symmetries:
$$ 
X_1=F_1(t)\partial_x,\qquad X_2=F_2(t)\partial_u,\qquad X_3=F_3(t)\partial_t,\qquad X_4=F_4(t)x^2\partial_u,\qquad X_5=F_5(t)x\partial_x,\qquad X_6=F_6(t)x\partial_u,\qquad X_7=F_7(t)u\partial_u,\qquad X_8=F_8(t)\left(\frac{1}{2}x^2\partial_x+xu\partial_u\right).
$$
The Lie symmetries of $u_{xxx}$ is different from the Kawaha KdV.

In another paper [[2]], the concept of hidden symmetries is raised. According to the paper, a *hidden symmetry* is a Lie point symmetry which appears in the target differential equation after a
change of order using a nonlocal transformation and which does not have a point counterpart
in the source equation.


  [1]: https://www.researchgate.net/publication/363703012_Differential_Invariants_of_the_Fifth-order_KdV_Types_of_Equations
  [2]: https://www.researchgate.net/publication/220535634_Hidden_and_Not_So_Hidden_Symmetries