Unable to create a map of a class in C++ - c++

Question

I'm not able to create a map of a certain class in C++.
Here is an example of what I'm talking about:
#include <iostream>
#include <map>
using namespace std;
class A {
private:
string name;
public:
A(){}
A(string name) : name(name){}
A(A& a) : name(a.name){}
~A(){}
string getName() { return name; }
};
int main() {
A a("a_name");
map<string, A> mapa;
mapa["a_name"] = a;
return 0;
}
When I compile the code, it gives me the following error:
In file included from /usr/include/c++/8/bits/stl_algobase.h:64,
from /usr/include/c++/8/bits/char_traits.h:39,
from /usr/include/c++/8/ios:40,
from /usr/include/c++/8/ostream:38,
from /usr/include/c++/8/iostream:39,
from prog.cpp:1:
/usr/include/c++/8/bits/stl_pair.h: In instantiation of ‘struct std::pair<const std::__cxx11::basic_string<char>, A>’:
/usr/include/c++/8/bits/stl_map.h:517:10: required from ‘std::map<_Key, _Tp, _Compare, _Alloc>::mapped_type& std::map<_Key, _Tp, _Compare, _Alloc>::operator[](std::map<_Key, _Tp, _Compare, _Alloc>::key_type&&) [with _Key = std::__cxx11::basic_string<char>; _Tp = A; _Compare = std::less<std::__cxx11::basic_string<char> >; _Alloc = std::allocator<std::pair<const std::__cxx11::basic_string<char>, A> >; std::map<_Key, _Tp, _Compare, _Alloc>::mapped_type = A; std::map<_Key, _Tp, _Compare, _Alloc>::key_type = std::__cxx11::basic_string<char>]’
prog.cpp:20:15: required from here
/usr/include/c++/8/bits/stl_pair.h:303:17: error: ‘constexpr std::pair<_T1, _T2>::pair(const std::pair<_T1, _T2>&) [with _T1 = const std::__cxx11::basic_string<char>; _T2 = A]’ declared to take const reference, but implicit declaration would take non-const
constexpr pair(const pair&) = default;
^~~~
Here is the code in ideone: https://www.ideone.com/sYsxcO
I need the map to save the object itself, and not just a pointer to it (in this case, the A objects).