diff --git a/src/config/config-set.spec.ts b/src/config/config-set.spec.ts
index 89af98723b..6cf028711c 100644
--- a/src/config/config-set.spec.ts
+++ b/src/config/config-set.spec.ts
@@ -415,7 +415,7 @@ describe('versions', () => {
   describe('with babel', () => {
     it('should return correct version map', () => {
       expect(createConfigSet({ tsJestConfig: { babelConfig: {} } }).versions).toEqual({
-        '@babel/core': '-',
+        '@babel/core': pkgVersion('@babel/core'),
         'babel-jest': pkgVersion('babel-jest'),
         jest: pkgVersion('jest'),
         'ts-jest': myModule.version,
diff --git a/src/config/create-jest-preset.spec.ts b/src/config/create-jest-preset.spec.ts
index bfadbd7e71..3347bb2f3d 100644
--- a/src/config/create-jest-preset.spec.ts
+++ b/src/config/create-jest-preset.spec.ts
@@ -25,18 +25,6 @@ it('should return correct defaults when allowJs is true', () => {
 it('should be able to use a base config', () => {
   expect(createJestPreset(undefined, {})).toMatchInlineSnapshot(`
 Object {
-  "moduleFileExtensions": Array [
-    "js",
-    "json",
-    "jsx",
-    "ts",
-    "tsx",
-    "node",
-  ],
-  "testMatch": Array [
-    "**/__tests__/**/*.[jt]s?(x)",
-    "**/?(*.)+(spec|test).[tj]s?(x)",
-  ],
   "transform": Object {
     "^.+\\\\.tsx?$": "ts-jest",
   },
diff --git a/src/config/create-jest-preset.ts b/src/config/create-jest-preset.ts
index 6a1f855f9c..5f82022005 100644
--- a/src/config/create-jest-preset.ts
+++ b/src/config/create-jest-preset.ts
@@ -12,8 +12,8 @@ const defaults = jestConfigPkg.defaults
 
 export interface TsJestPresets {
   transform: Record<string, string>
-  testMatch: string[]
-  moduleFileExtensions: string[]
+  testMatch: string[] | undefined
+  moduleFileExtensions: string[] | undefined
 }
 
 export interface CreateJestPresetOptions {
@@ -31,7 +31,7 @@ export function createJestPreset(
       ...from.transform,
       [allowJs ? '^.+\\.[tj]sx?$' : '^.+\\.tsx?$']: 'ts-jest',
     },
-    testMatch: from.testMatch || [],
-    moduleFileExtensions: from.moduleFileExtensions || [],
+    testMatch: from.testMatch || undefined,
+    moduleFileExtensions: from.moduleFileExtensions || undefined,
   }
 }
diff --git a/src/util/importer.spec.ts b/src/util/importer.spec.ts
index 1fec177b94..d5d47f55e3 100644
--- a/src/util/importer.spec.ts
+++ b/src/util/importer.spec.ts
@@ -116,8 +116,8 @@ describe('babelCore', () => {
   })
   it('should fail with correct error message', () => {
     expect(() => new Importer().babelCore(fakers.importReason())).toThrowErrorMatchingInlineSnapshot(`
-"Unable to load any of these modules: \\"@babel/core\\". [[BECAUSE]]. To fix it:
-    • for Babel 7: \`npm i -D babel-jest @babel/core\` (or \`yarn add --dev babel-jest @babel/core\`)"
+"Unable to load the module \\"@babel/core\\". [[BECAUSE]] To fix it:
+    ↳ install \\"@babel/core\\": \`npm i -D @babel/core\` (or \`yarn add --dev @babel/core\`)"
 `)
   })
 })
diff --git a/src/util/importer.ts b/src/util/importer.ts
index 46aaf804c7..4cbb5dcd95 100644
--- a/src/util/importer.ts
+++ b/src/util/importer.ts
@@ -47,17 +47,7 @@ export class Importer {
   }
 
   babelCore(why: ImportReasons): TBabelCore {
-    // the bridge will choose correct babel version
-    return this._import(why, 'babel-core', {
-      alternatives: ['@babel/core'],
-      installTip: [
-        // as in https://github.com/facebook/jest/tree/master/packages/babel-jest
-        {
-          label: 'for Babel 7',
-          module: `babel-jest @babel/core`,
-        },
-      ],
-    })
+    return this._import(why, '@babel/core')
   }
 
   typescript(why: ImportReasons, which: string): TTypeScript {